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Statistics: random number generators, PDF and CDF functions, and hypothesis tests. The module also includes some basic statistical functions such as mean, variance, skew, and etc.

Randomisation functions
val shuffle : 'a array -> 'a array

shuffle x return a new array of the shuffled x.

val choose : 'a array -> int -> 'a array

choose x n draw n samples from x without replecement.

val sample : 'a array -> int -> 'a array

sample x n draw n samples from x with replacement.

Basic statistical functions
val sum : float array -> float

sum x returns the summation of the elements in x.

val mean : float array -> float

mean x returns the mean of the elements in x.

val var : ?mean:float -> float array -> float

var x returns the variance of elements in x.

val std : ?mean:float -> float array -> float

std x calculates the standard deviation of x.

val sem : ?mean:float -> float array -> float

sem x calculates the standard error of x, also referred to as standard error of the mean.

val absdev : ?mean:float -> float array -> float

absdev x calculates the average absolute deviation of x.

val skew : ?mean:float -> ?sd:float -> float array -> float

skew x calculates the skewness (the third standardized moment) of x.

val kurtosis : ?mean:float -> ?sd:float -> float array -> float

kurtosis x calculates the Pearson's kurtosis of x, i.e. the fourth standardized moment of x.

val central_moment : int -> float array -> float

central_moment n x calculates the n th central moment of x.

val cov : ?m0:float -> ?m1:float -> float array -> float array -> float

cov x0 x1 calculates the covariance of x0 and x1, the mean of x0 and x1 can be specified by m0 and m1 respectively.

val concordant : 'a array -> 'b array -> int

concordant x y calculates the number of concordant pairs in the given arrays x and y. A pair of observations \((x_i, y_i)\) and \((x_j, y_j)\) is concordant if both elements in one pair are either greater than or less than both elements in the other pair, i.e., \((x_i > x_j) \land (y_i > y_j)\) or \((x_i < x_j) \land (y_i < y_j)\).

The function returns the count of such pairs.

  • parameter x

    The first array of observations.

  • parameter y

    The second array of observations.

  • returns

    The number of concordant pairs.

val discordant : 'a array -> 'b array -> int

discordant x y calculates the number of discordant pairs in the given arrays x and y. A pair of observations \((x_i, y_i)\) and \((x_j, y_j)\) is discordant if one element in one pair is greater than the corresponding element in the other pair, and the other element is smaller, i.e., \((x_i > x_j) \land (y_i < y_j)\) or \((x_i < x_j) \land (y_i > y_j)\).

The function returns the count of such pairs.

  • parameter x

    The first array of observations.

  • parameter y

    The second array of observations.

  • returns

    The number of discordant pairs.

val corrcoef : float array -> float array -> float

corrcoef x y calculates the Pearson correlation of x and y. Namely, corrcoef x y = cov(x, y) / (sigma_x * sigma_y).

val kendall_tau : float array -> float array -> float

kendall_tau x y calculates the Kendall Tau correlation between x and y.

val spearman_rho : float array -> float array -> float

spearman_rho x y calculates the Spearman Rho correlation between x and y.

val autocorrelation : ?lag:int -> float array -> float

autocorrelation ~lag x calculates the autocorrelation of x with the given lag.

val percentile : float array -> float -> float

percentile x p returns the p percentile of the data x. p is between 0. and 100. x does not need to be sorted beforehand.

val quantile : float array -> float -> float

quantile x p returns the p quantile of the data x. x does not need to be sorted beforehand. When computing several quantiles on the same data, it is more efficient to "pre-apply" the function, as in ``let f = quantile x in List.map f 0.1 ; 0.5 ``, in which case the data is sorted only once.

Raises Invalid_argument if p is not between 0 and 1.

val first_quartile : float array -> float

first_quartile x returns the first quartile of x, i.e. 25 percentiles.

val third_quartile : float array -> float

third_quartile x returns the third quartile of x, i.e. 75 percentiles.

val interquartile : float array -> float

interquartile x returns the interquartile range of x which is defined as the first quartile subtracted from the third quartile.

val median : float array -> float

median x returns the median of x.

val min : float array -> float

min x returns the minimum element in x.

val max : float array -> float

max x returns the maximum element in x.

val minmax : float array -> float * float

minmax x returns both (minimum, maximum) elements in x.

val min_i : float array -> int

min_i x returns the index of the minimum in x.

val max_i : float array -> int

max_i x returns the index of the maximum in x.

val minmax_i : float array -> int * int

minmax_i x returns the indices of both minimum and maximum in x.

val sort : ?inc:bool -> float array -> float array

sort x sorts the elements in the x in increasing order if inc = true, otherwise in decreasing order if inc=false. By default, inc is true. Note a copy is returned, the original data is not modified.

val argsort : ?inc:bool -> float array -> int array

argsort x sorts the elements in x and returns the indices mapping of the elements in the current array to their original position in x.

The sorting is in increasing order if inc = true, otherwise in decreasing order if inc=false. By default, inc is true.

val rank : ?ties_strategy:[ `Average | `Min | `Max ] -> float array -> float array

Computes sample's ranks.

The ranking order is from the smallest one to the largest. For example rank [|54.; 74.; 55.; 86.; 56.|] returns [|1.; 4.; 2.; 5.; 3.|]. Note that the ranking starts with one!

ties_strategy controls which ranks are assigned to equal values:

  • Average the mean of ranks should be assigned to each value. Default.
  • Min the minimum of ranks is assigned to each value.
  • Max the maximum of ranks is assigned to each value.
type histogram = Owl_base_stats.histogram

Type for computed histograms, with optional weighted counts and normalized counts.

val histogram : [ `Bins of float array | `N of int ] -> ?weights:float array -> float array -> histogram

histogram bins x creates a histogram from values in x. If bins matches `N n it will construct n equally spaced bins from the minimum to the maximum in x. If bins matches `Bins b, b is taken as the sorted array of boundaries of adjacent bin intervals. Bin boundaries are taken as left-inclusive, right-exclusive, except for the last bin which is also right-inclusive. Values outside the bins are dropped silently.

histogram bins ~weights x creates a weighted histogram with the given weights which must match x in length. The bare counts are also provided.

Returns a histogram including the n+1 bin boundaries, n counts and weighted counts if applicable, but without normalisation.

val histogram_sorted : [ `Bins of float array | `N of int ] -> ?weights:float array -> float array -> histogram

histogram_sorted bins x is like histogram but assumes that x is sorted already. This increases efficiency if there are less bins than data. Undefined results if x is not in fact sorted.

val normalise : histogram -> histogram

normalize hist calculates a probability mass function using hist.weighted_counts if present, otherwise using hist.counts. The result is stored in the normalised_counts field and sums to one.

val normalise_density : histogram -> histogram

normalize_density hist calculates a probability density function using hist.weighted_counts if present, otherwise using hist.counts. The result is normalized as a density that is piecewise constant over the bin intervals. That is, the sum over density times corresponding bin width is one. If bins are infinitely wide, their density is 0 and the sum over width times density of all finite bins is the total weight in the finite bins. The result is stored in the density field.

val pp_hist : Format.formatter -> histogram -> unit

Pretty-print summary information on a histogram record

val ecdf : float array -> float array * float array

ecdf x returns (x',f) which are the empirical cumulative distribution function f of x at points x'. x' is just x sorted in increasing order with duplicates removed. The function does not support nan values in the array x.

val z_score : mu:float -> sigma:float -> float array -> float array

z_score x calculates the z score of a given array x.

val t_score : float array -> float array

t_score x calculates the t score of a given array x.

val normalise_pdf : float array -> float array

normalise_pdf arr takes an array of floats arr representing a probability density function (PDF) and returns a new array where the values are normalized so that the sum of the array equals 1.

  • parameter arr

    The input array representing the unnormalized PDF.

  • returns

    A new array of the same length as arr, with values normalized to ensure the sum is 1.

val tukey_fences : ?k:float -> float array -> float * float

tukey_fences ?k x returns a tuple of the lower and upper boundaries for values that are not outliers. k defaults to the standard coefficient of 1.5. For first and third quartiles Q1 and `Q3`, the range is computed as follows:

  (Q1 - k*(Q3-Q1), Q3 + k*(Q3-Q1))
val gaussian_kde : ?bandwidth:[ `Silverman | `Scott ] -> ?n_points:int -> float array -> float array * float array

gaussian_kde x is a Gaussian kernel density estimator. The estimation of the pdf runs in `O(sample_size * n_points)`, and returns an array tuple (a, b) where a is a uniformly spaced points from the sample range at which the density function was estimated, and b is the estimates at these points.

Bandwidth selection rules is as follows: * Silverman: use `rule-of-thumb` for choosing the bandwidth. It defaults to 0.9 * min(SD, IQR / 1.34) * n^-0.2. * Scott: same as Silverman, but with a factor, equal to 1.06.

The default bandwidth value is Scott.

MCMC: Markov Chain Monte Carlo
val metropolis_hastings : (float array -> float) -> float array -> int -> float array array

metropolis_hastings target_density initial_state num_samples performs the Metropolis-Hastings algorithm to generate samples from a target distribution.

The Metropolis-Hastings algorithm is a Markov Chain Monte Carlo (MCMC) method that generates a sequence of samples from a probability distribution for which direct sampling is difficult. The algorithm uses a proposal distribution to explore the state space, accepting or rejecting proposed moves based on the ratio of the target densities.

  • parameter target_density

    A function that computes the density (up to a normalizing constant) of the target distribution at a given point.

  • parameter initial_state

    The starting point of the Markov chain, represented as an array of floats.

  • parameter num_samples

    The number of samples to generate.

  • returns

    A 2D array where each row represents a sample generated by the Metropolis-Hastings algorithm.

val gibbs_sampling : (float array -> int -> float) -> float array -> int -> float array array

gibbs_sampling conditional_sampler initial_state num_samples performs Gibbs sampling to generate samples from a multivariate distribution.

Gibbs sampling is a Markov Chain Monte Carlo (MCMC) algorithm used to generate samples from a joint distribution when direct sampling is difficult. It works by iteratively sampling each variable from its conditional distribution, given the current values of all other variables.

  • parameter conditional_sampler

    A function that takes the current state of the variables (as a float array) and the index of the variable to sample, and returns a new value for that variable sampled from its conditional distribution.

  • parameter initial_state

    The starting point of the Markov chain, represented as an array of floats.

  • parameter num_samples

    The number of samples to generate.

  • returns

    A 2D array where each row represents a sample generated by the Gibbs sampling algorithm.

Hypothesis tests
type hypothesis = {
  1. reject : bool;
  2. p_value : float;
  3. score : float;
}

Record type contains the result of a hypothesis test.

type tail =
  1. | BothSide
  2. | RightSide
  3. | LeftSide
    (*

    Types of alternative hypothesis tests: one-side, left-side, or right-side.

    *)
val pp_hypothesis : Format.formatter -> hypothesis -> unit

Pretty printer of hypothesis type

val z_test : mu:float -> sigma:float -> ?alpha:float -> ?side:tail -> float array -> hypothesis

z_test ~mu ~sigma ~alpha ~side x returns a test decision for the null hypothesis that the data x comes from a normal distribution with mean mu and a standard deviation sigma, using the z-test of alpha significance level. The alternative hypothesis is that the mean is not mu.

The result (h,p,z) : h is true if the test rejects the null hypothesis at the alpha significance level, and false otherwise. p is the p-value and z is the z-score.

val t_test : mu:float -> ?alpha:float -> ?side:tail -> float array -> hypothesis

t_test ~mu ~alpha ~side x returns a test decision of one-sample t-test which is a parametric test of the location parameter when the population standard deviation is unknown. mu is population mean, alpha is the significance level.

val t_test_paired : ?alpha:float -> ?side:tail -> float array -> float array -> hypothesis

t_test_paired ~alpha ~side x y returns a test decision for the null hypothesis that the data in x – y comes from a normal distribution with mean equal to zero and unknown variance, using the paired-sample t-test.

val t_test_unpaired : ?alpha:float -> ?side:tail -> ?equal_var:bool -> float array -> float array -> hypothesis

t_test_unpaired ~alpha ~side ~equal_var x y returns a test decision for the null hypothesis that the data in vectors x and y comes from independent random samples from normal distributions with equal means and equal but unknown variances, using the two-sample t-test. The alternative hypothesis is that the data in x and y comes from populations with unequal means.

equal_var indicates whether two samples have the same variance. If the two variances are not the same, the test is referred to as Welche's t-test.

val ks_test : ?alpha:float -> float array -> (float -> float) -> hypothesis

ks_test ~alpha x f returns a test decision for the null hypothesis that the data in vector x comes from independent random samples of the distribution with CDF f. The alternative hypothesis is that the data in x comes from a different distribution.

The result (h,p,d) : h is true if the test rejects the null hypothesis at the alpha significance level, and false otherwise. p is the p-value and d is the Kolmogorov-Smirnov test statistic.

val ks2_test : ?alpha:float -> float array -> float array -> hypothesis

ks2_test ~alpha x y returns a test decision for the null hypothesis that the data in vectors x and y come from independent random samples of the same distribution. The alternative hypothesis is that the data in x and y are sampled from different distributions.

The result (h,p,d): h is true if the test rejects the null hypothesis at the alpha significance level, and false otherwise. p is the p-value and d is the Kolmogorov-Smirnov test statistic.

val var_test : ?alpha:float -> ?side:tail -> variance:float -> float array -> hypothesis

var_test ~alpha ~side ~variance x returns a test decision for the null hypothesis that the data in x comes from a normal distribution with input variance, using the chi-square variance test. The alternative hypothesis is that x comes from a normal distribution with a different variance.

val jb_test : ?alpha:float -> float array -> hypothesis

jb_test ~alpha x returns a test decision for the null hypothesis that the data x comes from a normal distribution with an unknown mean and variance, using the Jarque-Bera test.

val fisher_test : ?alpha:float -> ?side:tail -> int -> int -> int -> int -> hypothesis

fisher_test ~alpha ~side a b c d fisher's exact test for contingency table | a, b | | c, d |

The result (h,p,z) : h is true if the test rejects the null hypothesis at the alpha significance level, and false otherwise. p is the p-value and z is prior odds ratio.

val runs_test : ?alpha:float -> ?side:tail -> ?v:float -> float array -> hypothesis

runs_test ~alpha ~v x returns a test decision for the null hypothesis that the data x comes in random order, against the alternative that they do not, by running Wald–Wolfowitz runs test. The test is based on the number of runs of consecutive values above or below the mean of x. ~v is the reference value, the default value is the median of x.

val mannwhitneyu : ?alpha:float -> ?side:tail -> float array -> float array -> hypothesis

mannwhitneyu ~alpha ~side x y Computes the Mann-Whitney rank test on samples x and y. If length of each sample less than 10 and no ties, then using exact test (see paper Ying Kuen Cheung and Jerome H. Klotz (1997) The Mann Whitney Wilcoxon distribution using linked list Statistica Sinica 7 805-813), else usning asymptotic normal distribution.

val wilcoxon : ?alpha:float -> ?side:tail -> float array -> float array -> hypothesis

wilcoxon ?alpha ?side x y performs the Wilcoxon signed-rank test on the paired samples x and y.

The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used to compare two related samples, matched samples, or repeated measurements on a single sample to assess whether their population mean ranks differ.

  • parameter alpha

    The significance level for the test, typically set to 0.05 by default. This parameter is optional.

  • parameter side

    Specifies the alternative hypothesis. It determines whether the test is one-sided or two-sided. The default is usually a two-sided test.

  • parameter x

    The first array of sample data.

  • parameter y

    The second array of sample data.

  • returns

    A hypothesis type, indicating whether to reject the null hypothesis or not based on the test.

Discrete random variables

The _rvs functions generate random numbers according to the specified distribution. _pdf are "density" functions that return the probability of the element specified by the arguments, while _cdf functions are cumulative distribution functions that return the probability of all elements less than or equal to the chosen element, and _sf functions are survival functions returning one minus the corresponding CDF function. `log` versions of functions return the result for the natural logarithm of a chosen element.

val uniform_int_rvs : a:int -> b:int -> int

uniform_rvs ~a ~b returns a random uniformly distributed integer between a and b, inclusive.

val binomial_rvs : p:float -> n:int -> int

binomial_rvs p n returns a random integer representing the number of successes in n trials with probability of success p on each trial.

val binomial_pdf : int -> p:float -> n:int -> float

binomial_pdf k ~p ~n returns the binomially distributed probability of k successes in n trials with probability p of success on each trial.

val binomial_logpdf : int -> p:float -> n:int -> float

binomial_logpdf k ~p ~n returns the log-binomially distributed probability of k successes in n trials with probability p of success on each trial.

val binomial_cdf : int -> p:float -> n:int -> float

binomial_cdf k ~p ~n returns the binomially distributed cumulative probability of less than or equal to k successes in n trials, with probability p on each trial.

val binomial_logcdf : int -> p:float -> n:int -> float

binomial_logcdf k ~p ~n returns the log-binomially distributed cumulative probability of less than or equal to k successes in n trials, with probability p on each trial.

val binomial_sf : int -> p:float -> n:int -> float

binomial_sf k ~p ~n is the binomial survival function, i.e. 1 - (binomial_cdf k ~p ~n).

val binomial_logsf : int -> p:float -> n:int -> float

binomial_loggf k ~p ~n is the logbinomial survival function, i.e. 1 - (binomial_logcdf k ~p ~n).

val hypergeometric_rvs : good:int -> bad:int -> sample:int -> int

hypergeometric_rvs ~good ~bad ~sample returns a random hypergeometrically distributed integer representing the number of successes in a sample (without replacement) of size ~sample from a population with ~good successful elements and ~bad unsuccessful elements.

val hypergeometric_pdf : int -> good:int -> bad:int -> sample:int -> float

hypergeometric_pdf k ~good ~bad ~sample returns the hypergeometrically distributed probability of k successes in a sample (without replacement) of ~sample elements from a population containing ~good successful elements and ~bad unsuccessful ones.

val hypergeometric_logpdf : int -> good:int -> bad:int -> sample:int -> float

hypergeometric_logpdf k ~good ~bad ~sample returns a value equivalent to a log-transformed result from hypergeometric_pdf.

val multinomial_rvs : int -> p:float array -> int array

multinomial_rvs n ~p generates random numbers of multinomial distribution from n trials. The probability mass function is as follows.

  P(x) = \frac{n!}{{x_1}! \cdot\cdot\cdot {x_k}!} p_{1}^{x_1} \cdot\cdot\cdot p_{k}^{x_k}

p is the probability mass of k categories. The elements in p should all be positive (result is undefined if there are negative values), but they don't need to sum to 1: the result is the same whether or not p is normalized. For implementation details, refer to :cite:`davis1993computer`.

val multinomial_pdf : int array -> p:float array -> float

multinomial_rvs x ~p return the probability of x given the probability mass of a multinomial distribution.

val multinomial_logpdf : int array -> p:float array -> float

multinomial_rvs x ~p returns the logarithm probability of x given the probability mass of a multinomial distribution.

val categorical_rvs : float array -> int

categorical_rvs p returns the value of a random variable which follows the categorical distribution. This is equavalent to only one trial from multinomial_rvs function, so it is just a simple wrapping.

val std_uniform_rvs : unit -> float

Continuous random variables

std_uniform_rvs () generates a random variate from the standard uniform distribution over the interval [0, 1\).

  • returns

    A float representing a random sample from the standard uniform distribution.

val uniform_rvs : a:float -> b:float -> float

uniform_rvs ~a ~b generates a random variate from the uniform distribution over the interval [a, b\).

  • parameter a

    The lower bound of the interval.

  • parameter b

    The upper bound of the interval.

  • returns

    A float representing a random sample from the uniform distribution over [a, b\).

val uniform_pdf : float -> a:float -> b:float -> float

uniform_pdf x ~a ~b computes the probability density function (PDF) of the uniform distribution at the point x over the interval [a, b\).

  • parameter x

    The point at which to evaluate the PDF.

  • parameter a

    The lower bound of the interval.

  • parameter b

    The upper bound of the interval.

  • returns

    The PDF value at x.

val uniform_logpdf : float -> a:float -> b:float -> float

uniform_logpdf x ~a ~b computes the natural logarithm of the probability density function (log-PDF) of the uniform distribution at the point x over the interval [a, b\).

  • parameter x

    The point at which to evaluate the log-PDF.

  • parameter a

    The lower bound of the interval.

  • parameter b

    The upper bound of the interval.

  • returns

    The log-PDF value at x.

val uniform_cdf : float -> a:float -> b:float -> float

uniform_cdf x ~a ~b computes the cumulative distribution function (CDF) of the uniform distribution at the point x over the interval [a, b\).

  • parameter x

    The point at which to evaluate the CDF.

  • parameter a

    The lower bound of the interval.

  • parameter b

    The upper bound of the interval.

  • returns

    The CDF value at x.

val uniform_logcdf : float -> a:float -> b:float -> float

uniform_logcdf x ~a ~b computes the natural logarithm of the cumulative distribution function (log-CDF) of the uniform distribution at the point x over the interval [a, b\).

  • parameter x

    The point at which to evaluate the log-CDF.

  • parameter a

    The lower bound of the interval.

  • parameter b

    The upper bound of the interval.

  • returns

    The log-CDF value at x.

val uniform_ppf : float -> a:float -> b:float -> float

uniform_ppf q ~a ~b computes the percent-point function (PPF), also known as the quantile function or inverse CDF, of the uniform distribution for a given probability q over the interval [a, b\).

  • parameter q

    The probability for which to compute the corresponding quantile.

  • parameter a

    The lower bound of the interval.

  • parameter b

    The upper bound of the interval.

  • returns

    The quantile corresponding to q.

val uniform_sf : float -> a:float -> b:float -> float

uniform_sf x ~a ~b computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the uniform distribution at the point x over the interval [a, b\).

  • parameter x

    The point at which to evaluate the SF.

  • parameter a

    The lower bound of the interval.

  • parameter b

    The upper bound of the interval.

  • returns

    The SF value at x.

val uniform_logsf : float -> a:float -> b:float -> float

uniform_logsf x ~a ~b computes the natural logarithm of the survival function (log-SF) of the uniform distribution at the point x over the interval [a, b\).

  • parameter x

    The point at which to evaluate the log-SF.

  • parameter a

    The lower bound of the interval.

  • parameter b

    The upper bound of the interval.

  • returns

    The log-SF value at x.

val uniform_isf : float -> a:float -> b:float -> float

uniform_isf q ~a ~b computes the inverse survival function (ISF), which is the inverse of the survival function (SF), of the uniform distribution for a given probability q over the interval [a, b\).

  • parameter q

    The probability for which to compute the corresponding value from the ISF.

  • parameter a

    The lower bound of the interval.

  • parameter b

    The upper bound of the interval.

  • returns

    The value corresponding to q from the ISF.

val exponential_rvs : lambda:float -> float

exponential_rvs ~lambda generates a random variate from the exponential distribution with rate parameter lambda.

  • parameter lambda

    The rate parameter of the exponential distribution (must be positive).

  • returns

    A float representing a random sample from the exponential distribution.

val exponential_pdf : float -> lambda:float -> float

exponential_pdf x ~lambda computes the probability density function (PDF) of the exponential distribution at the point x with rate parameter lambda.

  • parameter x

    The point at which to evaluate the PDF.

  • parameter lambda

    The rate parameter of the exponential distribution (must be positive).

  • returns

    The PDF value at x.

val exponential_logpdf : float -> lambda:float -> float

exponential_logpdf x ~lambda computes the natural logarithm of the probability density function (log-PDF) of the exponential distribution at the point x with rate parameter lambda.

  • parameter x

    The point at which to evaluate the log-PDF.

  • parameter lambda

    The rate parameter of the exponential distribution (must be positive).

  • returns

    The log-PDF value at x.

val exponential_cdf : float -> lambda:float -> float

exponential_cdf x ~lambda computes the cumulative distribution function (CDF) of the exponential distribution at the point x with rate parameter lambda.

  • parameter x

    The point at which to evaluate the CDF.

  • parameter lambda

    The rate parameter of the exponential distribution (must be positive).

  • returns

    The CDF value at x.

val exponential_logcdf : float -> lambda:float -> float

exponential_logcdf x ~lambda computes the natural logarithm of the cumulative distribution function (log-CDF) of the exponential distribution at the point x with rate parameter lambda.

  • parameter x

    The point at which to evaluate the log-CDF.

  • parameter lambda

    The rate parameter of the exponential distribution (must be positive).

  • returns

    The log-CDF value at x.

val exponential_ppf : float -> lambda:float -> float

exponential_ppf q ~lambda computes the percent-point function (PPF), also known as the quantile function or inverse CDF, of the exponential distribution for a given probability q with rate parameter lambda.

  • parameter q

    The probability for which to compute the corresponding quantile.

  • parameter lambda

    The rate parameter of the exponential distribution (must be positive).

  • returns

    The quantile corresponding to q.

val exponential_sf : float -> lambda:float -> float

exponential_sf x ~lambda computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the exponential distribution at the point x with rate parameter lambda.

  • parameter x

    The point at which to evaluate the SF.

  • parameter lambda

    The rate parameter of the exponential distribution (must be positive).

  • returns

    The SF value at x.

val exponential_logsf : float -> lambda:float -> float

exponential_logsf x ~lambda computes the natural logarithm of the survival function (log-SF) of the exponential distribution at the point x with rate parameter lambda.

  • parameter x

    The point at which to evaluate the log-SF.

  • parameter lambda

    The rate parameter of the exponential distribution (must be positive).

  • returns

    The log-SF value at x.

val exponential_isf : float -> lambda:float -> float

exponential_isf q ~lambda computes the inverse survival function (ISF), which is the inverse of the survival function (SF), of the exponential distribution for a given probability q with rate parameter lambda.

  • parameter q

    The probability for which to compute the corresponding value from the ISF.

  • parameter lambda

    The rate parameter of the exponential distribution (must be positive).

  • returns

    The value corresponding to q from the ISF.

val exponpow_rvs : a:float -> b:float -> float
  p(x) dx = (1/(2 a Gamma(1+1/b))) * exp(-|x/a|^b) dx
val exponpow_pdf : float -> a:float -> b:float -> float

exponpow_pdf x ~a ~b computes the probability density function (PDF) of the exponential power distribution at the point x with shape parameters a and b.

  • parameter x

    The point at which to evaluate the PDF.

  • parameter a

    The scale parameter of the distribution.

  • parameter b

    The shape parameter of the distribution.

  • returns

    The PDF value at x.

val exponpow_logpdf : float -> a:float -> b:float -> float

exponpow_logpdf x ~a ~b computes the natural logarithm of the probability density function (log-PDF) of the exponential power distribution at the point x with shape parameters a and b.

  • parameter x

    The point at which to evaluate the log-PDF.

  • parameter a

    The scale parameter of the distribution.

  • parameter b

    The shape parameter of the distribution.

  • returns

    The log-PDF value at x.

val exponpow_cdf : float -> a:float -> b:float -> float

exponpow_cdf x ~a ~b computes the cumulative distribution function (CDF) of the exponential power distribution at the point x with shape parameters a and b.

  • parameter x

    The point at which to evaluate the CDF.

  • parameter a

    The scale parameter of the distribution.

  • parameter b

    The shape parameter of the distribution.

  • returns

    The CDF value at x.

val exponpow_logcdf : float -> a:float -> b:float -> float

exponpow_logcdf x ~a ~b computes the natural logarithm of the cumulative distribution function (log-CDF) of the exponential power distribution at the point x with shape parameters a and b.

  • parameter x

    The point at which to evaluate the log-CDF.

  • parameter a

    The scale parameter of the distribution.

  • parameter b

    The shape parameter of the distribution.

  • returns

    The log-CDF value at x.

val exponpow_sf : float -> a:float -> b:float -> float

exponpow_sf x ~a ~b computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the exponential power distribution at the point x with shape parameters a and b.

  • parameter x

    The point at which to evaluate the SF.

  • parameter a

    The scale parameter of the distribution.

  • parameter b

    The shape parameter of the distribution.

  • returns

    The SF value at x.

val exponpow_logsf : float -> a:float -> b:float -> float

exponpow_logsf x ~a ~b computes the natural logarithm of the survival function (log-SF) of the exponential power distribution at the point x with shape parameters a and b.

  • parameter x

    The point at which to evaluate the log-SF.

  • parameter a

    The scale parameter of the distribution.

  • parameter b

    The shape parameter of the distribution.

  • returns

    The log-SF value at x.

val gaussian_rvs : mu:float -> sigma:float -> float

gaussian_rvs ~mu ~sigma generates a random variate from the Gaussian (normal) distribution with mean mu and standard deviation sigma.

  • parameter mu

    The mean of the distribution.

  • parameter sigma

    The standard deviation of the distribution.

  • returns

    A float representing a random sample from the Gaussian distribution.

val gaussian_pdf : float -> mu:float -> sigma:float -> float

gaussian_pdf x ~mu ~sigma computes the probability density function (PDF) of the Gaussian (normal) distribution at the point x with mean mu and standard deviation sigma.

  • parameter x

    The point at which to evaluate the PDF.

  • parameter mu

    The mean of the distribution.

  • parameter sigma

    The standard deviation of the distribution.

  • returns

    The PDF value at x.

val gaussian_logpdf : float -> mu:float -> sigma:float -> float

gaussian_logpdf x ~mu ~sigma computes the natural logarithm of the probability density function (log-PDF) of the Gaussian (normal) distribution at the point x with mean mu and standard deviation sigma.

  • parameter x

    The point at which to evaluate the log-PDF.

  • parameter mu

    The mean of the distribution.

  • parameter sigma

    The standard deviation of the distribution.

  • returns

    The log-PDF value at x.

val gaussian_cdf : float -> mu:float -> sigma:float -> float

gaussian_cdf x ~mu ~sigma computes the cumulative distribution function (CDF) of the Gaussian (normal) distribution at the point x with mean mu and standard deviation sigma.

  • parameter x

    The point at which to evaluate the CDF.

  • parameter mu

    The mean of the distribution.

  • parameter sigma

    The standard deviation of the distribution.

  • returns

    The CDF value at x.

val gaussian_logcdf : float -> mu:float -> sigma:float -> float

gaussian_logcdf x ~mu ~sigma computes the natural logarithm of the cumulative distribution function (log-CDF) of the Gaussian (normal) distribution at the point x with mean mu and standard deviation sigma.

  • parameter x

    The point at which to evaluate the log-CDF.

  • parameter mu

    The mean of the distribution.

  • parameter sigma

    The standard deviation of the distribution.

  • returns

    The log-CDF value at x.

val gaussian_ppf : float -> mu:float -> sigma:float -> float

gaussian_ppf q ~mu ~sigma computes the percent-point function (PPF), also known as the quantile function or inverse CDF, of the Gaussian (normal) distribution for a given probability q with mean mu and standard deviation sigma.

  • parameter q

    The probability for which to compute the corresponding quantile.

  • parameter mu

    The mean of the distribution.

  • parameter sigma

    The standard deviation of the distribution.

  • returns

    The quantile corresponding to q.

val gaussian_sf : float -> mu:float -> sigma:float -> float

gaussian_sf x ~mu ~sigma computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the Gaussian (normal) distribution at the point x with mean mu and standard deviation sigma.

  • parameter x

    The point at which to evaluate the SF.

  • parameter mu

    The mean of the distribution.

  • parameter sigma

    The standard deviation of the distribution.

  • returns

    The SF value at x.

val gaussian_logsf : float -> mu:float -> sigma:float -> float

gaussian_logsf x ~mu ~sigma computes the natural logarithm of the survival function (log-SF) of the Gaussian (normal) distribution at the point x with mean mu and standard deviation sigma.

  • parameter x

    The point at which to evaluate the log-SF.

  • parameter mu

    The mean of the distribution.

  • parameter sigma

    The standard deviation of the distribution.

  • returns

    The log-SF value at x.

val gaussian_isf : float -> mu:float -> sigma:float -> float

gaussian_isf q ~mu ~sigma computes the inverse survival function (ISF), which is the inverse of the survival function (SF), of the Gaussian (normal) distribution for a given probability q with mean mu and standard deviation sigma.

  • parameter q

    The probability for which to compute the corresponding value from the ISF.

  • parameter mu

    The mean of the distribution.

  • parameter sigma

    The standard deviation of the distribution.

  • returns

    The value corresponding to q from the ISF.

val gamma_rvs : shape:float -> scale:float -> float

gamma_rvs ~shape ~scale generates a random variate from the gamma distribution with shape parameter shape and scale parameter scale.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    A float representing a random sample from the gamma distribution.

val gamma_pdf : float -> shape:float -> scale:float -> float

gamma_pdf x ~shape ~scale computes the probability density function (PDF) of the gamma distribution at the point x with shape parameter shape and scale parameter scale.

  • parameter x

    The point at which to evaluate the PDF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The PDF value at x.

val gamma_logpdf : float -> shape:float -> scale:float -> float

gamma_logpdf x ~shape ~scale computes the natural logarithm of the probability density function (log-PDF) of the gamma distribution at the point x with shape parameter shape and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-PDF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-PDF value at x.

val gamma_cdf : float -> shape:float -> scale:float -> float

gamma_cdf x ~shape ~scale computes the cumulative distribution function (CDF) of the gamma distribution at the point x with shape parameter shape and scale parameter scale.

  • parameter x

    The point at which to evaluate the CDF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The CDF value at x.

val gamma_logcdf : float -> shape:float -> scale:float -> float

gamma_logcdf x ~shape ~scale computes the natural logarithm of the cumulative distribution function (log-CDF) of the gamma distribution at the point x with shape parameter shape and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-CDF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-CDF value at x.

val gamma_ppf : float -> shape:float -> scale:float -> float

gamma_ppf q ~shape ~scale computes the percent-point function (PPF), also known as the quantile function or inverse CDF, of the gamma distribution for a given probability q with shape parameter shape and scale parameter scale.

  • parameter q

    The probability for which to compute the corresponding quantile.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The quantile corresponding to q.

val gamma_sf : float -> shape:float -> scale:float -> float

gamma_sf x ~shape ~scale computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the gamma distribution at the point x with shape parameter shape and scale parameter scale.

  • parameter x

    The point at which to evaluate the SF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The SF value at x.

val gamma_logsf : float -> shape:float -> scale:float -> float

gamma_logsf x ~shape ~scale computes the natural logarithm of the survival function (log-SF) of the gamma distribution at the point x with shape parameter shape and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-SF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-SF value at x.

val gamma_isf : float -> shape:float -> scale:float -> float

gamma_isf q ~shape ~scale computes the inverse survival function (ISF), which is the inverse of the survival function (SF), of the gamma distribution for a given probability q with shape parameter shape and scale parameter scale.

  • parameter q

    The probability for which to compute the corresponding value from the ISF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The value corresponding to q from the ISF.

val beta_rvs : a:float -> b:float -> float

beta_rvs ~a ~b generates a random variate from the beta distribution with shape parameters a and b.

  • parameter a

    The first shape parameter of the distribution.

  • parameter b

    The second shape parameter of the distribution.

  • returns

    A float representing a random sample from the beta distribution.

val beta_pdf : float -> a:float -> b:float -> float

beta_pdf x ~a ~b computes the probability density function (PDF) of the beta distribution at the point x with shape parameters a and b.

  • parameter x

    The point at which to evaluate the PDF.

  • parameter a

    The first shape parameter of the distribution.

  • parameter b

    The second shape parameter of the distribution.

  • returns

    The PDF value at x.

val beta_logpdf : float -> a:float -> b:float -> float

beta_logpdf x ~a ~b computes the natural logarithm of the probability density function (log-PDF) of the beta distribution at the point x with shape parameters a and b.

  • parameter x

    The point at which to evaluate the log-PDF.

  • parameter a

    The first shape parameter of the distribution.

  • parameter b

    The second shape parameter of the distribution.

  • returns

    The log-PDF value at x.

val beta_cdf : float -> a:float -> b:float -> float

beta_cdf x ~a ~b computes the cumulative distribution function (CDF) of the beta distribution at the point x with shape parameters a and b.

  • parameter x

    The point at which to evaluate the CDF.

  • parameter a

    The first shape parameter of the distribution.

  • parameter b

    The second shape parameter of the distribution.

  • returns

    The CDF value at x.

val beta_logcdf : float -> a:float -> b:float -> float

beta_logcdf x ~a ~b computes the natural logarithm of the cumulative distribution function (log-CDF) of the beta distribution at the point x with shape parameters a and b.

  • parameter x

    The point at which to evaluate the log-CDF.

  • parameter a

    The first shape parameter of the distribution.

  • parameter b

    The second shape parameter of the distribution.

  • returns

    The log-CDF value at x.

val beta_ppf : float -> a:float -> b:float -> float

beta_ppf q ~a ~b computes the percent-point function (PPF), also known as the quantile function or inverse CDF, of the beta distribution for a given probability q with shape parameters a and b.

  • parameter q

    The probability for which to compute the corresponding quantile.

  • parameter a

    The first shape parameter of the distribution.

  • parameter b

    The second shape parameter of the distribution.

  • returns

    The quantile corresponding to q.

val beta_sf : float -> a:float -> b:float -> float

beta_sf x ~a ~b computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the beta distribution at the point x with shape parameters a and b.

  • parameter x

    The point at which to evaluate the SF.

  • parameter a

    The first shape parameter of the distribution.

  • parameter b

    The second shape parameter of the distribution.

  • returns

    The SF value at x.

val beta_logsf : float -> a:float -> b:float -> float

beta_logsf x ~a ~b computes the natural logarithm of the survival function (log-SF) of the beta distribution at the point x with shape parameters a and b.

  • parameter x

    The point at which to evaluate the log-SF.

  • parameter a

    The first shape parameter of the distribution.

  • parameter b

    The second shape parameter of the distribution.

  • returns

    The log-SF value at x.

val beta_isf : float -> a:float -> b:float -> float

beta_isf q ~a ~b computes the inverse survival function (ISF), which is the inverse of the survival function (SF), of the beta distribution for a given probability q with shape parameters a and b.

  • parameter q

    The probability for which to compute the corresponding value from the ISF.

  • parameter a

    The first shape parameter of the distribution.

  • parameter b

    The second shape parameter of the distribution.

  • returns

    The value corresponding to q from the ISF.

val chi2_rvs : df:float -> float

chi2_rvs ~df generates a random variate from the chi-square distribution with degrees of freedom df.

  • parameter df

    The degrees of freedom of the distribution.

  • returns

    A float representing a random sample from the chi-square distribution.

val chi2_pdf : float -> df:float -> float

chi2_pdf x ~df computes the probability density function (PDF) of the chi-square distribution at the point x with degrees of freedom df.

  • parameter x

    The point at which to evaluate the PDF.

  • parameter df

    The degrees of freedom of the distribution.

  • returns

    The PDF value at x.

val chi2_logpdf : float -> df:float -> float

chi2_logpdf x ~df computes the natural logarithm of the probability density function (log-PDF) of the chi-square distribution at the point x with degrees of freedom df.

  • parameter x

    The point at which to evaluate the log-PDF.

  • parameter df

    The degrees of freedom of the distribution.

  • returns

    The log-PDF value at x.

val chi2_cdf : float -> df:float -> float

chi2_cdf x ~df computes the cumulative distribution function (CDF) of the chi-square distribution at the point x with degrees of freedom df.

  • parameter x

    The point at which to evaluate the CDF.

  • parameter df

    The degrees of freedom of the distribution.

  • returns

    The CDF value at x.

val chi2_logcdf : float -> df:float -> float

chi2_logcdf x ~df computes the natural logarithm of the cumulative distribution function (log-CDF) of the chi-square distribution at the point x with degrees of freedom df.

  • parameter x

    The point at which to evaluate the log-CDF.

  • parameter df

    The degrees of freedom of the distribution.

  • returns

    The log-CDF value at x.

val chi2_ppf : float -> df:float -> float

chi2_ppf q ~df computes the percent-point function (PPF), also known as the quantile function or inverse CDF, of the chi-square distribution for a given probability q with degrees of freedom df.

  • parameter q

    The probability for which to compute the corresponding quantile.

  • parameter df

    The degrees of freedom of the distribution.

  • returns

    The quantile corresponding to q.

val chi2_sf : float -> df:float -> float

chi2_sf x ~df computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the chi-square distribution at the point x with degrees of freedom df.

  • parameter x

    The point at which to evaluate the SF.

  • parameter df

    The degrees of freedom of the distribution.

  • returns

    The SF value at x.

val chi2_logsf : float -> df:float -> float

chi2_logsf x ~df computes the natural logarithm of the survival function (log-SF) of the chi-square distribution at the point x with degrees of freedom df.

  • parameter x

    The point at which to evaluate the log-SF.

  • parameter df

    The degrees of freedom of the distribution.

  • returns

    The log-SF value at x.

val chi2_isf : float -> df:float -> float

chi2_isf q ~df computes the inverse survival function (ISF), which is the inverse of the survival function (SF), of the chi-square distribution for a given probability q with degrees of freedom df.

  • parameter q

    The probability for which to compute the corresponding value from the ISF.

  • parameter df

    The degrees of freedom of the distribution.

  • returns

    The value corresponding to q from the ISF.

val f_rvs : dfnum:float -> dfden:float -> float

f_rvs ~dfnum ~dfden generates a random variate from the F-distribution with numerator degrees of freedom dfnum and denominator degrees of freedom dfden.

The F-distribution is commonly used in the analysis of variance (ANOVA) and in the comparison of two variances.

  • parameter dfnum

    The numerator degrees of freedom.

  • parameter dfden

    The denominator degrees of freedom.

  • returns

    A float representing a random sample from the F-distribution.

val f_pdf : float -> dfnum:float -> dfden:float -> float

f_pdf x ~dfnum ~dfden computes the probability density function (PDF) of the F-distribution at the point x with numerator degrees of freedom dfnum and denominator degrees of freedom dfden.

  • parameter x

    The point at which to evaluate the PDF.

  • parameter dfnum

    The numerator degrees of freedom.

  • parameter dfden

    The denominator degrees of freedom.

  • returns

    The PDF value at x.

val f_logpdf : float -> dfnum:float -> dfden:float -> float

f_logpdf x ~dfnum ~dfden computes the natural logarithm of the probability density function (log-PDF) of the F-distribution at the point x with numerator degrees of freedom dfnum and denominator degrees of freedom dfden.

  • parameter x

    The point at which to evaluate the log-PDF.

  • parameter dfnum

    The numerator degrees of freedom.

  • parameter dfden

    The denominator degrees of freedom.

  • returns

    The log-PDF value at x.

val f_cdf : float -> dfnum:float -> dfden:float -> float

f_cdf x ~dfnum ~dfden computes the cumulative distribution function (CDF) of the F-distribution at the point x with numerator degrees of freedom dfnum and denominator degrees of freedom dfden.

  • parameter x

    The point at which to evaluate the CDF.

  • parameter dfnum

    The numerator degrees of freedom.

  • parameter dfden

    The denominator degrees of freedom.

  • returns

    The CDF value at x.

val f_logcdf : float -> dfnum:float -> dfden:float -> float

f_logcdf x ~dfnum ~dfden computes the natural logarithm of the cumulative distribution function (log-CDF) of the F-distribution at the point x with numerator degrees of freedom dfnum and denominator degrees of freedom dfden.

  • parameter x

    The point at which to evaluate the log-CDF.

  • parameter dfnum

    The numerator degrees of freedom.

  • parameter dfden

    The denominator degrees of freedom.

  • returns

    The log-CDF value at x.

val f_ppf : float -> dfnum:float -> dfden:float -> float

f_ppf q ~dfnum ~dfden computes the percent-point function (PPF), also known as the quantile function or inverse CDF, of the F-distribution for a given probability q with numerator degrees of freedom dfnum and denominator degrees of freedom dfden.

  • parameter q

    The probability for which to compute the corresponding quantile.

  • parameter dfnum

    The numerator degrees of freedom.

  • parameter dfden

    The denominator degrees of freedom.

  • returns

    The quantile corresponding to q.

val f_sf : float -> dfnum:float -> dfden:float -> float

f_sf x ~dfnum ~dfden computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the F-distribution at the point x with numerator degrees of freedom dfnum and denominator degrees of freedom dfden.

  • parameter x

    The point at which to evaluate the SF.

  • parameter dfnum

    The numerator degrees of freedom.

  • parameter dfden

    The denominator degrees of freedom.

  • returns

    The SF value at x.

val f_logsf : float -> dfnum:float -> dfden:float -> float

f_logsf x ~dfnum ~dfden computes the natural logarithm of the survival function (log-SF) of the F-distribution at the point x with numerator degrees of freedom dfnum and denominator degrees of freedom dfden.

  • parameter x

    The point at which to evaluate the log-SF.

  • parameter dfnum

    The numerator degrees of freedom.

  • parameter dfden

    The denominator degrees of freedom.

  • returns

    The log-SF value at x.

val f_isf : float -> dfnum:float -> dfden:float -> float

f_isf q ~dfnum ~dfden computes the inverse survival function (ISF), which is the inverse of the survival function (SF), of the F-distribution for a given probability q with numerator degrees of freedom dfnum and denominator degrees of freedom dfden.

  • parameter q

    The probability for which to compute the corresponding value from the ISF.

  • parameter dfnum

    The numerator degrees of freedom.

  • parameter dfden

    The denominator degrees of freedom.

  • returns

    The value corresponding to q from the ISF.

val cauchy_rvs : loc:float -> scale:float -> float

cauchy_rvs ~loc ~scale generates a random variate from the Cauchy distribution with location parameter loc and scale parameter scale.

The Cauchy distribution is a continuous probability distribution with heavy tails, often used in robust statistical methods.

  • parameter loc

    The location parameter of the distribution (the peak of the curve).

  • parameter scale

    The scale parameter of the distribution (half-width at half-maximum).

  • returns

    A float representing a random sample from the Cauchy distribution.

val cauchy_pdf : float -> loc:float -> scale:float -> float

cauchy_pdf x ~loc ~scale computes the probability density function (PDF) of the Cauchy distribution at the point x with location parameter loc and scale parameter scale.

  • parameter x

    The point at which to evaluate the PDF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The PDF value at x.

val cauchy_logpdf : float -> loc:float -> scale:float -> float

cauchy_logpdf x ~loc ~scale computes the natural logarithm of the probability density function (log-PDF) of the Cauchy distribution at the point x with location parameter loc and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-PDF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-PDF value at x.

val cauchy_cdf : float -> loc:float -> scale:float -> float

cauchy_cdf x ~loc ~scale computes the cumulative distribution function (CDF) of the Cauchy distribution at the point x with location parameter loc and scale parameter scale.

  • parameter x

    The point at which to evaluate the CDF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The CDF value at x.

val cauchy_logcdf : float -> loc:float -> scale:float -> float

cauchy_logcdf x ~loc ~scale computes the natural logarithm of the cumulative distribution function (log-CDF) of the Cauchy distribution at the point x with location parameter loc and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-CDF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-CDF value at x.

val cauchy_ppf : float -> loc:float -> scale:float -> float

cauchy_ppf q ~loc ~scale computes the percent-point function (PPF), also known as the quantile function or inverse CDF, of the Cauchy distribution for a given probability q with location parameter loc and scale parameter scale.

  • parameter q

    The probability for which to compute the corresponding quantile.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The quantile corresponding to q.

val cauchy_sf : float -> loc:float -> scale:float -> float

cauchy_sf x ~loc ~scale computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the Cauchy distribution at the point x with location parameter loc and scale parameter scale.

  • parameter x

    The point at which to evaluate the SF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The SF value at x.

val cauchy_logsf : float -> loc:float -> scale:float -> float

cauchy_logsf x ~loc ~scale computes the natural logarithm of the survival function (log-SF) of the Cauchy distribution at the point x with location parameter loc and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-SF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-SF value at x.

val cauchy_isf : float -> loc:float -> scale:float -> float

cauchy_isf q ~loc ~scale computes the inverse survival function (ISF), which is the inverse of the survival function (SF), of the Cauchy distribution for a given probability q with location parameter loc and scale parameter scale.

  • parameter q

    The probability for which to compute the corresponding value from the ISF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The value corresponding to q from the ISF.

val t_rvs : df:float -> loc:float -> scale:float -> float

t_rvs ~df ~loc ~scale generates a random variate from the Student's t-distribution with df degrees of freedom, location parameter loc, and scale parameter scale.

The Student's t-distribution is commonly used in statistics for small sample sizes or when the population standard deviation is unknown.

  • parameter df

    The degrees of freedom of the distribution.

  • parameter loc

    The location parameter of the distribution (the mean).

  • parameter scale

    The scale parameter of the distribution (the standard deviation).

  • returns

    A float representing a random sample from the t-distribution.

val t_pdf : float -> df:float -> loc:float -> scale:float -> float

t_pdf x ~df ~loc ~scale computes the probability density function (PDF) of the Student's t-distribution at the point x with df degrees of freedom, location parameter loc, and scale parameter scale.

  • parameter x

    The point at which to evaluate the PDF.

  • parameter df

    The degrees of freedom of the distribution.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The PDF value at x.

val t_logpdf : float -> df:float -> loc:float -> scale:float -> float

t_logpdf x ~df ~loc ~scale computes the natural logarithm of the probability density function (log-PDF) of the Student's t-distribution at the point x with df degrees of freedom, location parameter loc, and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-PDF.

  • parameter df

    The degrees of freedom of the distribution.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-PDF value at x.

val t_cdf : float -> df:float -> loc:float -> scale:float -> float

t_cdf x ~df ~loc ~scale computes the cumulative distribution function (CDF) of the Student's t-distribution at the point x with df degrees of freedom, location parameter loc, and scale parameter scale.

  • parameter x

    The point at which to evaluate the CDF.

  • parameter df

    The degrees of freedom of the distribution.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The CDF value at x.

val t_logcdf : float -> df:float -> loc:float -> scale:float -> float

t_logcdf x ~df ~loc ~scale computes the natural logarithm of the cumulative distribution function (log-CDF) of the Student's t-distribution at the point x with df degrees of freedom, location parameter loc, and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-CDF.

  • parameter df

    The degrees of freedom of the distribution.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-CDF value at x.

val t_ppf : float -> df:float -> loc:float -> scale:float -> float

t_ppf q ~df ~loc ~scale computes the percent-point function (PPF), also known as the quantile function or inverse CDF, of the Student's t-distribution for a given probability q with df degrees of freedom, location parameter loc, and scale parameter scale.

  • parameter q

    The probability for which to compute the corresponding quantile.

  • parameter df

    The degrees of freedom of the distribution.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The quantile corresponding to q.

val t_sf : float -> df:float -> loc:float -> scale:float -> float

t_sf x ~df ~loc ~scale computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the Student's t-distribution at the point x with df degrees of freedom, location parameter loc, and scale parameter scale.

  • parameter x

    The point at which to evaluate the SF.

  • parameter df

    The degrees of freedom of the distribution.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The SF value at x.

val t_logsf : float -> df:float -> loc:float -> scale:float -> float

t_logsf x ~df ~loc ~scale computes the natural logarithm of the survival function (log-SF) of the Student's t-distribution at the point x with df degrees of freedom, location parameter loc, and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-SF.

  • parameter df

    The degrees of freedom of the distribution.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-SF value at x.

val t_isf : float -> df:float -> loc:float -> scale:float -> float

t_isf q ~df ~loc ~scale computes the inverse survival function (ISF), which is the inverse of the survival function (SF), of the Student's t-distribution for a given probability q with df degrees of freedom, location parameter loc, and scale parameter scale.

  • parameter q

    The probability for which to compute the corresponding value from the ISF.

  • parameter df

    The degrees of freedom of the distribution.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The value corresponding to q from the ISF.

val vonmises_rvs : mu:float -> kappa:float -> float

vonmises_rvs ~mu ~kappa generates a random variate from the von Mises distribution with mean direction mu and concentration parameter kappa.

The von Mises distribution is often used as a circular analogue of the normal distribution for data measured in angles or on a circle.

  • parameter mu

    The mean direction of the distribution.

  • parameter kappa

    The concentration parameter of the distribution, where larger values indicate higher concentration around the mean direction.

  • returns

    A float representing a random sample from the von Mises distribution.

val vonmises_pdf : float -> mu:float -> kappa:float -> float

vonmises_pdf x ~mu ~kappa computes the probability density function (PDF) of the von Mises distribution at the point x with mean direction mu and concentration parameter kappa.

  • parameter x

    The point at which to evaluate the PDF.

  • parameter mu

    The mean direction of the distribution.

  • parameter kappa

    The concentration parameter of the distribution.

  • returns

    The PDF value at x.

val vonmises_logpdf : float -> mu:float -> kappa:float -> float

vonmises_logpdf x ~mu ~kappa computes the natural logarithm of the probability density function (log-PDF) of the von Mises distribution at the point x with mean direction mu and concentration parameter kappa.

  • parameter x

    The point at which to evaluate the log-PDF.

  • parameter mu

    The mean direction of the distribution.

  • parameter kappa

    The concentration parameter of the distribution.

  • returns

    The log-PDF value at x.

val vonmises_cdf : float -> mu:float -> kappa:float -> float

vonmises_cdf x ~mu ~kappa computes the cumulative distribution function (CDF) of the von Mises distribution at the point x with mean direction mu and concentration parameter kappa.

  • parameter x

    The point at which to evaluate the CDF.

  • parameter mu

    The mean direction of the distribution.

  • parameter kappa

    The concentration parameter of the distribution.

  • returns

    The CDF value at x.

val vonmises_logcdf : float -> mu:float -> kappa:float -> float

vonmises_logcdf x ~mu ~kappa computes the natural logarithm of the cumulative distribution function (log-CDF) of the von Mises distribution at the point x with mean direction mu and concentration parameter kappa.

  • parameter x

    The point at which to evaluate the log-CDF.

  • parameter mu

    The mean direction of the distribution.

  • parameter kappa

    The concentration parameter of the distribution.

  • returns

    The log-CDF value at x.

val vonmises_sf : float -> mu:float -> kappa:float -> float

vonmises_sf x ~mu ~kappa computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the von Mises distribution at the point x with mean direction mu and concentration parameter kappa.

  • parameter x

    The point at which to evaluate the SF.

  • parameter mu

    The mean direction of the distribution.

  • parameter kappa

    The concentration parameter of the distribution.

  • returns

    The SF value at x.

val vonmises_logsf : float -> mu:float -> kappa:float -> float

vonmises_logsf x ~mu ~kappa computes the natural logarithm of the survival function (log-SF) of the von Mises distribution at the point x with mean direction mu and concentration parameter kappa.

  • parameter x

    The point at which to evaluate the log-SF.

  • parameter mu

    The mean direction of the distribution.

  • parameter kappa

    The concentration parameter of the distribution.

  • returns

    The log-SF value at x.

val lomax_rvs : shape:float -> scale:float -> float

lomax_rvs ~shape ~scale generates a random variate from the Lomax distribution, also known as the Pareto distribution of the second kind, with shape parameter shape and scale parameter scale.

The Lomax distribution is often used in survival analysis and heavy-tailed modeling.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    A float representing a random sample from the Lomax distribution.

val lomax_pdf : float -> shape:float -> scale:float -> float

lomax_pdf x ~shape ~scale computes the probability density function (PDF) of the Lomax distribution at the point x with shape parameter shape and scale parameter scale.

  • parameter x

    The point at which to evaluate the PDF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The PDF value at x.

val lomax_logpdf : float -> shape:float -> scale:float -> float

lomax_logpdf x ~shape ~scale computes the natural logarithm of the probability density function (log-PDF) of the Lomax distribution at the point x with shape parameter shape and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-PDF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-PDF value at x.

val lomax_cdf : float -> shape:float -> scale:float -> float

lomax_cdf x ~shape ~scale computes the cumulative distribution function (CDF) of the Lomax distribution at the point x with shape parameter shape and scale parameter scale.

  • parameter x

    The point at which to evaluate the CDF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The CDF value at x.

val lomax_logcdf : float -> shape:float -> scale:float -> float

lomax_logcdf x ~shape ~scale computes the natural logarithm of the cumulative distribution function (log-CDF) of the Lomax distribution at the point x with shape parameter shape and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-CDF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-CDF value at x.

val lomax_ppf : float -> shape:float -> scale:float -> float

lomax_ppf q ~shape ~scale computes the percent-point function (PPF), also known as the quantile function or inverse CDF, of the Lomax distribution for a given probability q with shape parameter shape and scale parameter scale.

  • parameter q

    The probability for which to compute the corresponding quantile.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The quantile corresponding to q.

val lomax_sf : float -> shape:float -> scale:float -> float

lomax_sf x ~shape ~scale computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the Lomax distribution at the point x with shape parameter shape and scale parameter scale.

  • parameter x

    The point at which to evaluate the SF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The SF value at x.

val lomax_logsf : float -> shape:float -> scale:float -> float

lomax_logsf x ~shape ~scale computes the natural logarithm of the survival function (log-SF) of the Lomax distribution at the point x with shape parameter shape and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-SF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-SF value at x.

val lomax_isf : float -> shape:float -> scale:float -> float

lomax_isf q ~shape ~scale computes the inverse survival function (ISF), which is the inverse of the survival function (SF), of the Lomax distribution for a given probability q with shape parameter shape and scale parameter scale.

  • parameter q

    The probability for which to compute the corresponding value from the ISF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The value corresponding to q from the ISF.

val weibull_rvs : shape:float -> scale:float -> float

weibull_rvs ~shape ~scale generates a random variate from the Weibull distribution with shape parameter shape and scale parameter scale.

The Weibull distribution is commonly used in reliability analysis and modeling life data.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    A float representing a random sample from the Weibull distribution.

val weibull_pdf : float -> shape:float -> scale:float -> float

weibull_pdf x ~shape ~scale computes the probability density function (PDF) of the Weibull distribution at the point x with shape parameter shape and scale parameter scale.

  • parameter x

    The point at which to evaluate the PDF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The PDF value at x.

val weibull_logpdf : float -> shape:float -> scale:float -> float

weibull_logpdf x ~shape ~scale computes the natural logarithm of the probability density function (log-PDF) of the Weibull distribution at the point x with shape parameter shape and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-PDF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-PDF value at x.

val weibull_cdf : float -> shape:float -> scale:float -> float

weibull_cdf x ~shape ~scale computes the cumulative distribution function (CDF) of the Weibull distribution at the point x with shape parameter shape and scale parameter scale.

  • parameter x

    The point at which to evaluate the CDF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The CDF value at x.

val weibull_logcdf : float -> shape:float -> scale:float -> float

weibull_logcdf x ~shape ~scale computes the natural logarithm of the cumulative distribution function (log-CDF) of the Weibull distribution at the point x with shape parameter shape and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-CDF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-CDF value at x.

val weibull_ppf : float -> shape:float -> scale:float -> float

weibull_ppf q ~shape ~scale computes the percent-point function (PPF), also known as the quantile function or inverse CDF, of the Weibull distribution for a given probability q with shape parameter shape and scale parameter scale.

  • parameter q

    The probability for which to compute the corresponding quantile.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The quantile corresponding to q.

val weibull_sf : float -> shape:float -> scale:float -> float

weibull_sf x ~shape ~scale computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the Weibull distribution at the point x with shape parameter shape and scale parameter scale.

  • parameter x

    The point at which to evaluate the SF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The SF value at x.

val weibull_logsf : float -> shape:float -> scale:float -> float

weibull_logsf x ~shape ~scale computes the natural logarithm of the survival function (log-SF) of the Weibull distribution at the point x with shape parameter shape and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-SF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-SF value at x.

val weibull_isf : float -> shape:float -> scale:float -> float

weibull_isf q ~shape ~scale computes the inverse survival function (ISF), which is the inverse of the survival function (SF), of the Weibull distribution for a given probability q with shape parameter shape and scale parameter scale.

  • parameter q

    The probability for which to compute the corresponding value from the ISF.

  • parameter shape

    The shape parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The value corresponding to q from the ISF.

val laplace_rvs : loc:float -> scale:float -> float

laplace_rvs ~loc ~scale generates a random variate from the Laplace distribution, also known as the double exponential distribution, with location parameter loc and scale parameter scale.

The Laplace distribution is often used in statistical methods for detecting outliers.

  • parameter loc

    The location parameter of the distribution (the mean).

  • parameter scale

    The scale parameter of the distribution (the standard deviation).

  • returns

    A float representing a random sample from the Laplace distribution.

val laplace_pdf : float -> loc:float -> scale:float -> float

laplace_pdf x ~loc ~scale computes the probability density function (PDF) of the Laplace distribution at the point x with location parameter loc and scale parameter scale.

  • parameter x

    The point at which to evaluate the PDF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The PDF value at x.

val laplace_logpdf : float -> loc:float -> scale:float -> float

laplace_logpdf x ~loc ~scale computes the natural logarithm of the probability density function (log-PDF) of the Laplace distribution at the point x with location parameter loc and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-PDF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-PDF value at x.

val laplace_cdf : float -> loc:float -> scale:float -> float

laplace_cdf x ~loc ~scale computes the cumulative distribution function (CDF) of the Laplace distribution at the point x with location parameter loc and scale parameter scale.

  • parameter x

    The point at which to evaluate the CDF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The CDF value at x.

val laplace_logcdf : float -> loc:float -> scale:float -> float

laplace_logcdf x ~loc ~scale computes the natural logarithm of the cumulative distribution function (log-CDF) of the Laplace distribution at the point x with location parameter loc and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-CDF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-CDF value at x.

val laplace_ppf : float -> loc:float -> scale:float -> float

laplace_ppf q ~loc ~scale computes the percent-point function (PPF), also known as the quantile function or inverse CDF, of the Laplace distribution for a given probability q with location parameter loc and scale parameter scale.

  • parameter q

    The probability for which to compute the corresponding quantile.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The quantile corresponding to q.

val laplace_sf : float -> loc:float -> scale:float -> float

laplace_sf x ~loc ~scale computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the Laplace distribution at the point x with location parameter loc and scale parameter scale.

  • parameter x

    The point at which to evaluate the SF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The SF value at x.

val laplace_logsf : float -> loc:float -> scale:float -> float

laplace_logsf x ~loc ~scale computes the natural logarithm of the survival function (log-SF) of the Laplace distribution at the point x with location parameter loc and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-SF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-SF value at x.

val laplace_isf : float -> loc:float -> scale:float -> float

laplace_isf q ~loc ~scale computes the inverse survival function (ISF), which is the inverse of the survival function (SF), of the Laplace distribution for a given probability q with location parameter loc and scale parameter scale.

  • parameter q

    The probability for which to compute the corresponding value from the ISF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The value corresponding to q from the ISF.

val gumbel1_rvs : a:float -> b:float -> float

gumbel1_rvs ~a ~b generates a random variate from the Gumbel Type I distribution with location parameter a and scale parameter b.

The Gumbel distribution is commonly used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions.

  • parameter a

    The location parameter of the distribution.

  • parameter b

    The scale parameter of the distribution.

  • returns

    A float representing a random sample from the Gumbel Type I distribution.

val gumbel1_pdf : float -> a:float -> b:float -> float

gumbel1_pdf x ~a ~b computes the probability density function (PDF) of the Gumbel Type I distribution at the point x with location parameter a and scale parameter b.

  • parameter x

    The point at which to evaluate the PDF.

  • parameter a

    The location parameter of the distribution.

  • parameter b

    The scale parameter of the distribution.

  • returns

    The PDF value at x.

val gumbel1_logpdf : float -> a:float -> b:float -> float

gumbel1_logpdf x ~a ~b computes the natural logarithm of the probability density function (log-PDF) of the Gumbel Type I distribution at the point x with location parameter a and scale parameter b.

  • parameter x

    The point at which to evaluate the log-PDF.

  • parameter a

    The location parameter of the distribution.

  • parameter b

    The scale parameter of the distribution.

  • returns

    The log-PDF value at x.

val gumbel1_cdf : float -> a:float -> b:float -> float

gumbel1_cdf x ~a ~b computes the cumulative distribution function (CDF) of the Gumbel Type I distribution at the point x with location parameter a and scale parameter b.

  • parameter x

    The point at which to evaluate the CDF.

  • parameter a

    The location parameter of the distribution.

  • parameter b

    The scale parameter of the distribution.

  • returns

    The CDF value at x.

val gumbel1_logcdf : float -> a:float -> b:float -> float

gumbel1_logcdf x ~a ~b computes the natural logarithm of the cumulative distribution function (log-CDF) of the Gumbel Type I distribution at the point x with location parameter a and scale parameter b.

  • parameter x

    The point at which to evaluate the log-CDF.

  • parameter a

    The location parameter of the distribution.

  • parameter b

    The scale parameter of the distribution.

  • returns

    The log-CDF value at x.

val gumbel1_ppf : float -> a:float -> b:float -> float

gumbel1_ppf q ~a ~b computes the percent-point function (PPF), also known as the quantile function or inverse CDF, of the Gumbel Type I distribution for a given probability q with location parameter a and scale parameter b.

  • parameter q

    The probability for which to compute the corresponding quantile.

  • parameter a

    The location parameter of the distribution.

  • parameter b

    The scale parameter of the distribution.

  • returns

    The quantile corresponding to q.

val gumbel1_sf : float -> a:float -> b:float -> float

gumbel1_sf x ~a ~b computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the Gumbel Type I distribution at the point x with location parameter a and scale parameter b.

  • parameter x

    The point at which to evaluate the SF.

  • parameter a

    The location parameter of the distribution.

  • parameter b

    The scale parameter of the distribution.

  • returns

    The SF value at x.

val gumbel1_logsf : float -> a:float -> b:float -> float

gumbel1_logsf x ~a ~b computes the natural logarithm of the survival function (log-SF) of the Gumbel Type I distribution at the point x with location parameter a and scale parameter b.

  • parameter x

    The point at which to evaluate the log-SF.

  • parameter a

    The location parameter of the distribution.

  • parameter b

    The scale parameter of the distribution.

  • returns

    The log-SF value at x.

val gumbel1_isf : float -> a:float -> b:float -> float

gumbel1_isf q ~a ~b computes the inverse survival function (ISF), which is the inverse of the survival function (SF), of the Gumbel Type I distribution for a given probability q with location parameter a and scale parameter b.

  • parameter q

    The probability for which to compute the corresponding value from the ISF.

  • parameter a

    The location parameter of the distribution.

  • parameter b

    The scale parameter of the distribution.

  • returns

    The value corresponding to q from the ISF.

val gumbel2_rvs : a:float -> b:float -> float

gumbel2_rvs ~a ~b generates a random variate from the Gumbel Type II distribution with location parameter a and scale parameter b.

The Gumbel Type II distribution is used for modeling extreme values in various fields.

  • parameter a

    The location parameter of the distribution.

  • parameter b

    The scale parameter of the distribution.

  • returns

    A float representing a random sample from the Gumbel Type II distribution.

val gumbel2_pdf : float -> a:float -> b:float -> float

gumbel2_pdf x ~a ~b computes the probability density function (PDF) of the Gumbel Type II distribution at the point x with location parameter a and scale parameter b.

  • parameter x

    The point at which to evaluate the PDF.

  • parameter a

    The location parameter of the distribution.

  • parameter b

    The scale parameter of the distribution.

  • returns

    The PDF value at x.

val gumbel2_logpdf : float -> a:float -> b:float -> float

gumbel2_logpdf x ~a ~b computes the natural logarithm of the probability density function (log-PDF) of the Gumbel Type II distribution at the point x with location parameter a and scale parameter b.

  • parameter x

    The point at which to evaluate the log-PDF.

  • parameter a

    The location parameter of the distribution.

  • parameter b

    The scale parameter of the distribution.

  • returns

    The log-PDF value at x.

val gumbel2_cdf : float -> a:float -> b:float -> float

gumbel2_cdf x ~a ~b computes the cumulative distribution function (CDF) of the Gumbel Type II distribution at the point x with location parameter a and scale parameter b.

  • parameter x

    The point at which to evaluate the CDF.

  • parameter a

    The location parameter of the distribution.

  • parameter b

    The scale parameter of the distribution.

  • returns

    The CDF value at x.

val gumbel2_logcdf : float -> a:float -> b:float -> float

gumbel2_logcdf x ~a ~b computes the natural logarithm of the cumulative distribution function (log-CDF) of the Gumbel Type II distribution at the point x with location parameter a and scale parameter b.

  • parameter x

    The point at which to evaluate the log-CDF.

  • parameter a

    The location parameter of the distribution.

  • parameter b

    The scale parameter of the distribution.

  • returns

    The log-CDF value at x.

val gumbel2_ppf : float -> a:float -> b:float -> float

gumbel2_ppf q ~a ~b computes the percent-point function (PPF), also known as the quantile function or inverse CDF, of the Gumbel Type II distribution for a given probability q with location parameter a and scale parameter b.

  • parameter q

    The probability for which to compute the corresponding quantile.

  • parameter a

    The location parameter of the distribution.

  • parameter b

    The scale parameter of the distribution.

  • returns

    The quantile corresponding to q.

val gumbel2_sf : float -> a:float -> b:float -> float

gumbel2_sf x ~a ~b computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the Gumbel Type II distribution at the point x with location parameter a and scale parameter b.

  • parameter x

    The point at which to evaluate the SF.

  • parameter a

    The location parameter of the distribution.

  • parameter b

    The scale parameter of the distribution.

  • returns

    The SF value at x.

val gumbel2_logsf : float -> a:float -> b:float -> float

gumbel2_logsf x ~a ~b computes the natural logarithm of the survival function (log-SF) of the Gumbel Type II distribution at the point x with location parameter a and scale parameter b.

  • parameter x

    The point at which to evaluate the log-SF.

  • parameter a

    The location parameter of the distribution.

  • parameter b

    The scale parameter of the distribution.

  • returns

    The log-SF value at x.

val gumbel2_isf : float -> a:float -> b:float -> float

gumbel2_isf q ~a ~b computes the inverse survival function (ISF), which is the inverse of the survival function (SF), of the Gumbel Type II distribution for a given probability q with location parameter a and scale parameter b.

  • parameter q

    The probability for which to compute the corresponding value from the ISF.

  • parameter a

    The location parameter of the distribution.

  • parameter b

    The scale parameter of the distribution.

  • returns

    The value corresponding to q from the ISF.

val logistic_rvs : loc:float -> scale:float -> float

logistic_rvs ~loc ~scale generates a random variate from the logistic distribution with location parameter loc and scale parameter scale.

The logistic distribution is often used in logistic regression and as a growth model.

  • parameter loc

    The location parameter of the distribution (the mean).

  • parameter scale

    The scale parameter of the distribution (related to the standard deviation).

  • returns

    A float representing a random sample from the logistic distribution.

val logistic_pdf : float -> loc:float -> scale:float -> float

logistic_pdf x ~loc ~scale computes the probability density function (PDF) of the logistic distribution at the point x with location parameter loc and scale parameter scale.

  • parameter x

    The point at which to evaluate the PDF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The PDF value at x.

val logistic_logpdf : float -> loc:float -> scale:float -> float

logistic_logpdf x ~loc ~scale computes the natural logarithm of the probability density function (log-PDF) of the logistic distribution at the point x with location parameter loc and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-PDF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-PDF value at x.

val logistic_cdf : float -> loc:float -> scale:float -> float

logistic_cdf x ~loc ~scale computes the cumulative distribution function (CDF) of the logistic distribution at the point x with location parameter loc and scale parameter scale.

  • parameter x

    The point at which to evaluate the CDF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The CDF value at x.

val logistic_logcdf : float -> loc:float -> scale:float -> float

logistic_logcdf x ~loc ~scale computes the natural logarithm of the cumulative distribution function (log-CDF) of the logistic distribution at the point x with location parameter loc and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-CDF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-CDF value at x.

val logistic_ppf : float -> loc:float -> scale:float -> float

logistic_ppf q ~loc ~scale computes the percent-point function (PPF), also known as the quantile function or inverse CDF, of the logistic distribution for a given probability q with location parameter loc and scale parameter scale.

  • parameter q

    The probability for which to compute the corresponding quantile.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The quantile corresponding to q.

val logistic_sf : float -> loc:float -> scale:float -> float

logistic_sf x ~loc ~scale computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the logistic distribution at the point x with location parameter loc and scale parameter scale.

  • parameter x

    The point at which to evaluate the SF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The SF value at x.

val logistic_logsf : float -> loc:float -> scale:float -> float

logistic_logsf x ~loc ~scale computes the natural logarithm of the survival function (log-SF) of the logistic distribution at the point x with location parameter loc and scale parameter scale.

  • parameter x

    The point at which to evaluate the log-SF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The log-SF value at x.

val logistic_isf : float -> loc:float -> scale:float -> float

logistic_isf q ~loc ~scale computes the inverse survival function (ISF), which is the inverse of the survival function (SF), of the logistic distribution for a given probability q with location parameter loc and scale parameter scale.

  • parameter q

    The probability for which to compute the corresponding value from the ISF.

  • parameter loc

    The location parameter of the distribution.

  • parameter scale

    The scale parameter of the distribution.

  • returns

    The value corresponding to q from the ISF.

val lognormal_rvs : mu:float -> sigma:float -> float

lognormal_rvs ~mu ~sigma generates a random variate from the log-normal distribution with parameters mu (mean of the underlying normal distribution) and sigma (standard deviation of the underlying normal distribution).

The log-normal distribution is commonly used to model positive-valued data with a distribution that is skewed to the right.

  • parameter mu

    The mean of the underlying normal distribution.

  • parameter sigma

    The standard deviation of the underlying normal distribution.

  • returns

    A float representing a random sample from the log-normal distribution.

val lognormal_pdf : float -> mu:float -> sigma:float -> float

lognormal_pdf x ~mu ~sigma computes the probability density function (PDF) of the log-normal distribution at the point x with parameters mu (mean of the underlying normal distribution) and sigma (standard deviation of the underlying normal distribution).

  • parameter x

    The point at which to evaluate the PDF.

  • parameter mu

    The mean of the underlying normal distribution.

  • parameter sigma

    The standard deviation of the underlying normal distribution.

  • returns

    The PDF value at x.

val lognormal_logpdf : float -> mu:float -> sigma:float -> float

lognormal_logpdf x ~mu ~sigma computes the natural logarithm of the probability density function (log-PDF) of the log-normal distribution at the point x with parameters mu and sigma.

  • parameter x

    The point at which to evaluate the log-PDF.

  • parameter mu

    The mean of the underlying normal distribution.

  • parameter sigma

    The standard deviation of the underlying normal distribution.

  • returns

    The log-PDF value at x.

val lognormal_cdf : float -> mu:float -> sigma:float -> float

lognormal_cdf x ~mu ~sigma computes the cumulative distribution function (CDF) of the log-normal distribution at the point x with parameters mu and sigma.

  • parameter x

    The point at which to evaluate the CDF.

  • parameter mu

    The mean of the underlying normal distribution.

  • parameter sigma

    The standard deviation of the underlying normal distribution.

  • returns

    The CDF value at x.

val lognormal_logcdf : float -> mu:float -> sigma:float -> float

lognormal_logcdf x ~mu ~sigma computes the natural logarithm of the cumulative distribution function (log-CDF) of the log-normal distribution at the point x with parameters mu and sigma.

  • parameter x

    The point at which to evaluate the log-CDF.

  • parameter mu

    The mean of the underlying normal distribution.

  • parameter sigma

    The standard deviation of the underlying normal distribution.

  • returns

    The log-CDF value at x.

val lognormal_ppf : float -> mu:float -> sigma:float -> float

lognormal_ppf q ~mu ~sigma computes the percent-point function (PPF), also known as the quantile function or inverse CDF, of the log-normal distribution for a given probability q with parameters mu and sigma.

  • parameter q

    The probability for which to compute the corresponding quantile.

  • parameter mu

    The mean of the underlying normal distribution.

  • parameter sigma

    The standard deviation of the underlying normal distribution.

  • returns

    The quantile corresponding to q.

val lognormal_sf : float -> mu:float -> sigma:float -> float

lognormal_sf x ~mu ~sigma computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the log-normal distribution at the point x with parameters mu and sigma.

  • parameter x

    The point at which to evaluate the SF.

  • parameter mu

    The mean of the underlying normal distribution.

  • parameter sigma

    The standard deviation of the underlying normal distribution.

  • returns

    The SF value at x.

val lognormal_logsf : float -> mu:float -> sigma:float -> float

lognormal_logsf x ~mu ~sigma computes the natural logarithm of the survival function (log-SF) of the log-normal distribution at the point x with parameters mu and sigma.

  • parameter x

    The point at which to evaluate the log-SF.

  • parameter mu

    The mean of the underlying normal distribution.

  • parameter sigma

    The standard deviation of the underlying normal distribution.

  • returns

    The log-SF value at x.

val lognormal_isf : float -> mu:float -> sigma:float -> float

lognormal_isf q ~mu ~sigma computes the inverse survival function (ISF), which is the inverse of the survival function (SF), of the log-normal distribution for a given probability q with parameters mu and sigma.

  • parameter q

    The probability for which to compute the corresponding value from the ISF.

  • parameter mu

    The mean of the underlying normal distribution.

  • parameter sigma

    The standard deviation of the underlying normal distribution.

  • returns

    The value corresponding to q from the ISF.

val rayleigh_rvs : sigma:float -> float

rayleigh_rvs ~sigma generates a random variate from the Rayleigh distribution with scale parameter sigma.

The Rayleigh distribution is commonly used to model the magnitude of a vector in two-dimensional space with Gaussian-distributed components.

  • parameter sigma

    The scale parameter of the distribution.

  • returns

    A float representing a random sample from the Rayleigh distribution.

val rayleigh_pdf : float -> sigma:float -> float

rayleigh_pdf x ~sigma computes the probability density function (PDF) of the Rayleigh distribution at the point x with scale parameter sigma.

  • parameter x

    The point at which to evaluate the PDF.

  • parameter sigma

    The scale parameter of the distribution.

  • returns

    The PDF value at x.

val rayleigh_logpdf : float -> sigma:float -> float

rayleigh_logpdf x ~sigma computes the natural logarithm of the probability density function (log-PDF) of the Rayleigh distribution at the point x with scale parameter sigma.

  • parameter x

    The point at which to evaluate the log-PDF.

  • parameter sigma

    The scale parameter of the distribution.

  • returns

    The log-PDF value at x.

val rayleigh_cdf : float -> sigma:float -> float

rayleigh_cdf x ~sigma computes the cumulative distribution function (CDF) of the Rayleigh distribution at the point x with scale parameter sigma.

  • parameter x

    The point at which to evaluate the CDF.

  • parameter sigma

    The scale parameter of the distribution.

  • returns

    The CDF value at x.

val rayleigh_logcdf : float -> sigma:float -> float

rayleigh_logcdf x ~sigma computes the natural logarithm of the cumulative distribution function (log-CDF) of the Rayleigh distribution at the point x with scale parameter sigma.

  • parameter x

    The point at which to evaluate the log-CDF.

  • parameter sigma

    The scale parameter of the distribution.

  • returns

    The log-CDF value at x.

val rayleigh_ppf : float -> sigma:float -> float

rayleigh_ppf q ~sigma computes the percent-point function (PPF), also known as the quantile function or inverse CDF, of the Rayleigh distribution for a given probability q with scale parameter sigma.

  • parameter q

    The probability for which to compute the corresponding quantile.

  • parameter sigma

    The scale parameter of the distribution.

  • returns

    The quantile corresponding to q.

val rayleigh_sf : float -> sigma:float -> float

rayleigh_sf x ~sigma computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the Rayleigh distribution at the point x with scale parameter sigma.

  • parameter x

    The point at which to evaluate the SF.

  • parameter sigma

    The scale parameter of the distribution.

  • returns

    The SF value at x.

val rayleigh_logsf : float -> sigma:float -> float

rayleigh_logsf x ~sigma computes the natural logarithm of the survival function (log-SF) of the Rayleigh distribution at the point x with scale parameter sigma.

  • parameter x

    The point at which to evaluate the log-SF.

  • parameter sigma

    The scale parameter of the distribution.

  • returns

    The log-SF value at x.

val rayleigh_isf : float -> sigma:float -> float

rayleigh_isf q ~sigma computes the inverse survival function (ISF), which is the inverse of the survival function (SF), of the Rayleigh distribution for a given probability q with scale parameter sigma.

  • parameter q

    The probability for which to compute the corresponding value from the ISF.

  • parameter sigma

    The scale parameter of the distribution.

  • returns

    The value corresponding to q from the ISF.

val dirichlet_rvs : alpha:float array -> float array

dirichlet_rvs ~alpha returns random variables of K-1 order Dirichlet distribution, follows the following probability dense function.

f(x_1,...,x_K; \alpha_1,...,\alpha_K) = \frac{1}{\mathbf{B(\alpha)}} \prod_{i=1}^K x_i^{\alpha_i - 1}

The normalising constant is the multivariate Beta function, which can be expressed in terms of the gamma function:

\mathbf{B(\alpha)} = \frac{\prod_{i=1}^K \Gamma(\alpha_i)}{\Gamma(\sum_{i=1}^K \alpha_i)}

Note that x is a standard K-1 simplex, i.e. \sum_i^K x_i = 1 and x_i \ge 0, \forall x_i \in [1,K].

val dirichlet_pdf : float array -> alpha:float array -> float

dirichlet_pdf x ~alpha computes the probability density function (PDF) of the Dirichlet distribution for the input vector x with concentration parameters alpha.

The Dirichlet distribution is a multivariate generalization of the Beta distribution and is commonly used as a prior distribution in Bayesian statistics, particularly in the context of categorical data and multinomial distributions.

  • parameter x

    The input vector for which to evaluate the PDF, typically representing proportions that sum to 1.

  • parameter alpha

    The concentration parameters of the distribution, where each element must be greater than 0.

  • returns

    The PDF value for the given input vector x.

val dirichlet_logpdf : float array -> alpha:float array -> float

dirichlet_logpdf x ~alpha computes the natural logarithm of the probability density function (log-PDF) of the Dirichlet distribution for the input vector x with concentration parameters alpha.

  • parameter x

    The input vector for which to evaluate the log-PDF, typically representing proportions that sum to 1.

  • parameter alpha

    The concentration parameters of the distribution, where each element must be greater than 0.

  • returns

    The log-PDF value for the given input vector x.

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