Owl_stats (p.owl.0.3.7.doc.owl.Owl

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`` calcuates 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

TODO

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

TODO

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`` calcuates the Kendall Tau correlation between ``x`` and ``y``.

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

``spearman_rho x y`` calcuates the Spearman Rho correlation between ``x`` and ``y``.

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

``autocorrelation ~lag x`` calcuates 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``. ``p`` is between 0. and 1. ``x`` does not need to be sorted beforehand.

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 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.

val histogram : float array -> int -> int array

``histogram x n`` creates a histogram of ``n`` buckets for ``x``.

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.

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

``z_score x`` calcuates the z score of a given array ``x``.

val t_score : float array -> float array

``t_score x`` calcuates the t score of a given array ``x``.

val normlise_pdf : float array -> float array

TODO

MCMC: Markov Chain Monte Carlo

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

TODO: ``metropolis_hastings f p n`` is Metropolis-Hastings MCMC algorithm. f is pdf of the p

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

TODO: ``gibbs_sampling f p n`` is Gibbs sampler. f is a sampler based on the full conditional function of all variables

Hypothesis tests

type hypothesis = {

reject : bool;
p_value : float;
score : float;

}

Record type contains the result of a hypothesis test.

type tail =

| BothSide
| RightSide
| LeftSide
(*
Types of alternative hypothesis tests: one-side, left-side, or right-side.
*)

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 runnign 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

TODO

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``.

Continuous random variables

val std_uniform_rvs : unit -> float

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

val exponential_rvs : lambda:float -> float

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

val chi2_rvs : df:float -> float

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

val rayleigh_rvs : sigma:float -> float

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO

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

TODO