# Univariate Distributions¶

Univariate distributions are the distributions whose variate forms are Univariate (i.e each sample is a scalar). Abstract types for univariate distributions:

const UnivariateDistribution{S<:ValueSupport} = Distribution{Univariate,S}

const DiscreteUnivariateDistribution   = Distribution{Univariate, Discrete}
const ContinuousUnivariateDistribution = Distribution{Univariate, Continuous}


## Common Interface¶

A series of methods are implemented for each univariate distribution, which provide useful functionalities such as moment computation, pdf evaluation, and sampling (i.e. random number generation).

### Parameter Retrieval¶

params(d)

Return a tuple of parameters.

Note: Let d be a distribution of type D, then D(params(d)...) will construct exactly the same distribution as d.

succprob(d)

Get the probability of success.

failprob(d)

Get the probability of failure.

scale(d)

Get the scale parameter.

location(d)

Get the location parameter.

shape(d)

Get the shape parameter.

rate(d)

Get the rate parameter.

ncategories(d)

Get the number of categories.

ntrials(d)

Get the number of trials.

dof(d)

Get the degrees of freedom.

Note: params are defined for all univariate distributions, while other parameter retrieval methods are only defined for those distributions for which these parameters make sense. See below for details.

### Computation of statistics¶

Let d be a distribution:

mean(d)

Return the expectation of distribution d.

var(d)

Return the variance of distribution d.

std(d)

Return the standard deviation of distribution d, i.e. sqrt(var(d)).

median(d)

Return the median value of distribution d.

modes(d)

Return an array of all modes of d.

mode(d)

Return the mode of distribution d. If d has multiple modes, it returns the first one, i.e. modes(d)[1].

skewness(d)

Return the skewness of distribution d.

kurtosis(d)

Return the excess kurtosis of distribution d.

isplatykurtic(d)

Return whether d is platykurtic (i.e kurtosis(d) > 0).

isleptokurtic(d)

Return whether d is leptokurtic (i.e kurtosis(d) < 0).

ismesokurtic(d)

Return whether d is mesokurtic (i.e kurtosis(d) == 0).

entropy(d)

Return the entropy value of distribution d.

entropy(d, base)

Return the entropy value of distribution d, w.r.t. a given base.

mgf(d, t)

Evaluate the moment generating function of distribution d.

cf(d, t)

Evaluate the characteristic function of distribution d.

### Probability Evaluation¶

insupport(d, x)

When x is a scalar, it returns whether x is within the support of d. When x is an array, it returns whether every element in x is within the support of d.

pdf(d, x)

The pdf value(s) evaluated at x.

pdf(d, rgn)

Get/compute the probabilities over a range of values. Here, rgn should be in the form of a:b.

Note: computing the probabilities over a contiguous range of values can take advantage of the recursive relations between probability masses and thus is often more efficient than computing these probabilities individually.

pdf(d)

Get/compute the entire probability vector of d. This is equivalent to pdf(d, minimum(d):maximum(d)).

Note: this method is only defined for bounded distributions.

logpdf(d, x)

The logarithm of the pdf value(s) evaluated at x, i.e. log(pdf(x)).

Note: The internal implementation may directly evaluate logpdf instead of first computing pdf and then taking the logarithm, for better numerical stability or efficiency.

loglikelihood(d, x)

The log-likelihood of distribution d w.r.t. all samples contained in array x.

cdf(d, x)

The cumulative distribution function evaluated at x.

logcdf(d, x)

The logarithm of the cumulative function value(s) evaluated at x, i.e. log(cdf(x)).

ccdf(d, x)

The complementary cumulative function evaluated at x, i.e. 1 - cdf(d, x).

logccdf(d, x)

The logarithm of the complementary cumulative function values evaluated at x, i.e. log(ccdf(x)).

quantile(d, q)

The quantile value. Let x = quantile(d, q), then cdf(d, x) = q.

cquantile(d, q)

The complementary quantile value, i.e. quantile(d, 1-q).

invlogcdf(d, lp)

The inverse function of logcdf.

invlogccdf(d, lp)

The inverse function of logccdf.

### Vectorized evaluation¶

Vectorized computation and inplace vectorized computation are supported for the following functions:

• pdf
• logpdf
• cdf
• logcdf
• ccdf
• logccdf
• quantile
• cquantile
• invlogcdf
• invlogccdf

For example, when x is an array, then r = pdf(d, x) returns an array r of the same size, such that r[i] = pdf(d, x[i]). One can also use pdf! to write results to pre-allocated storage, as pdf!(r, d, x).

### Sampling (Random number generation)¶

rand(d)

Draw a sample from d

rand(d, n)

Return a vector comprised of n independent samples from the distribution d.

rand(d, dims)

Return an array of size dims that is filled with independent samples from the distribution d.

rand(d, dim0, dim1, ...)

Similar to rand(d, dims) above, except that the dimensions can be input as individual integers.

For example, one can write rand(d, 2, 3) or rand(d, (2, 3)), which are equivalent.

rand!(d, arr)

Fills a pre-allocated array arr with independent samples from the distribution d.