# Continuous Distributions

The functions described here are among the most commonly used continuous univariate statistical distributions. You can compute their densities, means, variances, and other related properties. The distributions themselves are represented in the symbolic form

. Functions such as

Mean, which give properties of statistical distributions, take the symbolic representation of the distribution as an argument.

"Discrete Distributions" describes many common discrete univariate statistical distributions.

Distributions related to the normal distribution.

The lognormal distribution

LogNormalDistribution is the distribution followed by the exponential of a normally distributed random variable. This distribution arises when many independent random variables are combined in a multiplicative fashion. The half-normal distribution

HalfNormalDistribution[] is proportional to the distribution

NormalDistribution[0, 1/( Sqrt[2/])] limited to the domain

.

The inverse Gaussian distribution

InverseGaussianDistribution, sometimes called the Wald distribution, is the distribution of first passage times in Brownian motion with positive drift.

Distributions related to normally distributed samples.

If

, ...,

are independent normal random variables with unit variance and mean zero, then

has a

distribution with

degrees of freedom. If a normal variable is standardized by subtracting its mean and dividing by its standard deviation, then the sum of squares of such quantities follows this distribution. The

distribution is most typically used when describing the variance of normal samples.

If

follows a

distribution with

degrees of freedom,

follows the

*inverse ** distribution* InverseChiSquareDistribution. A

*scaled inverse ** distribution* with

degrees of freedom and scale

can be given as

InverseChiSquareDistribution. Inverse

distributions are commonly used as prior distributions for the variance in Bayesian analysis of normally distributed samples.

A variable that has a Student

distribution can also be written as a function of normal random variables. Let

* *and

be independent random variables, where

is a standard normal distribution and

is a

variable with

degrees of freedom. In this case,

has a

distribution with

degrees of freedom. The Student

distribution is symmetric about the vertical axis, and characterizes the ratio of a normal variable to its standard deviation. Location and scale parameters can be included as

and

in

StudentTDistribution. When

, the

distribution is the same as the Cauchy distribution.

The

-ratio distribution is the distribution of the ratio of two independent

variables divided by their respective degrees of freedom. It is commonly used when comparing the variances of two populations in hypothesis testing.

Distributions that are derived from normal distributions with nonzero means are called

*noncentral distributions*.

The sum of the squares of

normally distributed random variables with variance

and nonzero means follows a noncentral

distribution

NoncentralChiSquareDistribution. The noncentrality parameter

is the sum of the squares of the means of the random variables in the sum. Note that in various places in the literature,

or

is used as the noncentrality parameter.

The noncentral Student

distribution

NoncentralStudentTDistribution describes the ratio

where

is a central

random variable with

degrees of freedom, and

is an independent normally distributed random variable with variance

and mean

.

The noncentral

-ratio distribution

NoncentralFRatioDistribution is the distribution of the ratio of

to

, where

is a noncentral

random variable with noncentrality parameter

and

degrees of freedom and

is a central

random variable with

degrees of freedom.

Piecewise linear distributions.

The triangular distribution

TriangularDistribution is a triangular distribution for

with maximum probability at

and

. If

is

,

TriangularDistribution is the symmetric triangular distribution

TriangularDistribution.

The uniform distribution

UniformDistribution, commonly referred to as the rectangular distribution, characterizes a random variable whose value is everywhere equally likely. An example of a uniformly distributed random variable is the location of a point chosen randomly on a line from

min to

max.

BetaDistribution[,] | continuous beta distribution with shape parameters and |

CauchyDistribution[a,b] | Cauchy distribution with location parameter a and scale parameter b |

ChiDistribution[] | distribution with degrees of freedom |

ExponentialDistribution[] | exponential distribution with scale inversely proportional to parameter |

ExtremeValueDistribution[,] | extreme maximum value (Fisher-Tippett) distribution with location parameter and scale parameter |

GammaDistribution[,] | gamma distribution with shape parameter and scale parameter |

GumbelDistribution[,] | Gumbel minimum extreme value distribution with location parameter and scale parameter |

InverseGammaDistribution[,] | inverse gamma distribution with shape parameter and scale parameter |

LaplaceDistribution[,] | Laplace (double exponential) distribution with mean and scale parameter |

LevyDistribution[,] | Lévy distribution with location parameter and dispersion parameter |

LogisticDistribution[,] | logistic distribution with mean and scale parameter |

MaxwellDistribution[] | Maxwell (Maxwell-Boltzmann) distribution with scale parameter |

ParetoDistribution[k,] | Pareto distribution with minimum value parameter k and shape parameter |

RayleighDistribution[] | Rayleigh distribution with scale parameter |

WeibullDistribution[,] | Weibull distribution with shape parameter and scale parameter |

Other continuous statistical distributions.

If

is uniformly distributed on

, then the random variable

follows a Cauchy distribution

CauchyDistribution, with

and

.

When

and

, the gamma distribution

GammaDistribution describes the distribution of a sum of squares of

-unit normal random variables. This form of the gamma distribution is called a

distribution with

degrees of freedom. When

, the gamma distribution takes on the form of the exponential distribution

ExponentialDistribution[], often used in describing the waiting time between events.

If a random variable

follows the gamma distribution

GammaDistribution,

follows the

*inverse gamma distribution* InverseGammaDistribution. If a random variable

follows

InverseGammaDistribution,

follows a

*Lévy distribution* LevyDistribution.

When

and

have independent gamma distributions with equal scale parameters, the random variable

follows the beta distribution

BetaDistribution, where

and

are the shape parameters of the gamma variables.

The

distribution

ChiDistribution[] is followed by the square root of a

random variable. For

, the

distribution is identical to

HalfNormalDistribution[] with

. For

, the

distribution is identical to the Rayleigh distribution

RayleighDistribution[] with

. For

, the

distribution is identical to the Maxwell-Boltzmann distribution

MaxwellDistribution[] with

.

The Laplace distribution

LaplaceDistribution is the distribution of the difference of two independent random variables with identical exponential distributions. The logistic distribution

LogisticDistribution is frequently used in place of the normal distribution when a distribution with longer tails is desired.

The Pareto distribution

ParetoDistribution may be used to describe income, with

representing the minimum income possible.

The Weibull distribution

WeibullDistribution is commonly used in engineering to describe the lifetime of an object. The extreme value distribution

ExtremeValueDistribution is the limiting distribution for the largest values in large samples drawn from a variety of distributions, including the normal distribution. The limiting distribution for the smallest values in such samples is the Gumbel distribution,

GumbelDistribution. The names "extreme value" and "Gumbel distribution" are sometimes used interchangeably because the distributions of the largest and smallest extreme values are related by a linear change of variable. The extreme value distribution is also sometimes referred to as the log-Weibull distribution because of logarithmic relationships between an extreme value-distributed random variable and a properly shifted and scaled Weibull-distributed random variable.

PDF[dist,x] | probability density function at x |

CDF[dist,x] | cumulative distribution function at x |

InverseCDF[dist,q] | the value of x such that CDF equals q |

Quantile[dist,q] | q quantile |

Mean[dist] | mean |

Variance[dist] | variance |

StandardDeviation[dist] | standard deviation |

Skewness[dist] | coefficient of skewness |

Kurtosis[dist] | coefficient of kurtosis |

CharacteristicFunction[dist,t] | characteristic function |

Expectation[f[x],xdist] | expectation of for x distributed according to dist |

Median[dist] | median |

Quartiles[dist] | list of the , , quantiles for dist |

InterquartileRange[dist] | difference between the first and third quartiles |

QuartileDeviation[dist] | half the interquartile range |

QuartileSkewness[dist] | quartile-based skewness measure |

RandomVariate[dist] | pseudorandom number with specified distribution |

RandomVariate[dist,dims] | pseudorandom array with dimensionality dims, and elements from the specified distribution |

Some functions of statistical distributions.

The preceding table gives a list of some of the more common functions available for distributions in

*Mathematica*.

The cumulative distribution function (CDF) at

is given by the integral of the probability density function (PDF) up to

. The PDF can therefore be obtained by differentiating the CDF (perhaps in a generalized sense). In this package the distributions are represented in symbolic form.

PDF evaluates the density at

if

is a numerical value, and otherwise leaves the function in symbolic form. Similarly,

CDF gives the cumulative distribution.

The inverse CDF

InverseCDF gives the value of

at which

CDF reaches

. The median is given by

InverseCDF. Quartiles, deciles, and percentiles are particular values of the inverse CDF. Quartile skewness is equivalent to

, where

,

, and

are the first, second, and third quartiles, respectively. Inverse CDFs are used in constructing confidence intervals for statistical parameters.

InverseCDF and

Quantile are equivalent for continuous distributions.

The mean

Mean[dist] is the expectation of the random variable distributed according to

dist and is usually denoted by

. The mean is given by

, where

is the PDF of the distribution. The variance

Variance[dist] is given by

. The square root of the variance is called the standard deviation, and is usually denoted by

.

The

Skewness[dist] and

Kurtosis[dist] functions give shape statistics summarizing the asymmetry and the peakedness of a distribution, respectively. Skewness is given by

and kurtosis is given by

.

The characteristic function

CharacteristicFunction is given by

. In the discrete case,

. Each distribution has a unique characteristic function, which is sometimes used instead of the PDF to define a distribution.

The expected value

Expectation of a function

g is given by

. In the discrete case, the expected value of

g is given by

.

RandomVariate[dist] gives pseudorandom numbers from the specified distribution.

This gives a symbolic representation of the gamma distribution with

and

.

Out[1]= | |

Here is the cumulative distribution function evaluated at

.

Out[2]= | |

This is the cumulative distribution function. It is given in terms of the built-in function

GammaRegularized.

Out[3]= | |

Here is a plot of the cumulative distribution function.

Out[4]= | |

This is a pseudorandom array with elements distributed according to the gamma distribution.

Out[5]= | |