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.
are independent normal random variables with unit variance and mean zero, then
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.
degrees of freedom,
follows the inverse distribution InverseChiSquareDistribution
. A scaled inverse distribution
degrees of freedom and scale
can be given as InverseChiSquareDistribution
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
be independent random variables, where
is a standard normal distribution and
degrees of freedom. In this case,
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
distribution is the same as the Cauchy distribution.
-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
. 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,
is used as the noncentrality parameter.
The noncentral Student
describes the ratio
is a central
random variable with
degrees of freedom, and
is an independent normally distributed random variable with variance
-ratio distribution NoncentralFRatioDistribution
is the distribution of the ratio of
is a noncentral
random variable with noncentrality parameter
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
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
|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.
is uniformly distributed on
, then the random variable
follows a Cauchy distribution CauchyDistribution
, 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
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 a Lévy distribution LevyDistribution
have independent gamma distributions with equal scale parameters, the random variable
follows the beta distribution BetaDistribution
are the shape parameters of the gamma variables.
is followed by the square root of a
random variable. For
distribution is identical to HalfNormalDistribution
distribution is identical to the Rayleigh distribution RayleighDistribution
distribution is identical to the Maxwell-Boltzmann distribution MaxwellDistribution
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 |
|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|
|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
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
. The median is given by InverseCDF
. Quartiles, deciles, and percentiles are particular values of the inverse CDF. Quartile skewness is equivalent to
are the first, second, and third quartiles, respectively. Inverse CDFs are used in constructing confidence intervals for statistical parameters. InverseCDF
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
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
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
gives pseudorandom numbers from the specified distribution.
This gives a symbolic representation of the gamma distribution with
Here is the cumulative distribution function evaluated at
This is the cumulative distribution function. It is given in terms of the built-in function GammaRegularized
Here is a plot of the cumulative distribution function.
This is a pseudorandom array with elements distributed according to the gamma distribution.