# 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.

NormalDistribution[,] | normal (Gaussian) distribution with mean and standard deviation |

HalfNormalDistribution[] | half-normal distribution with scale inversely proportional to parameter |

LogNormalDistribution[,] | lognormal distribution based on a normal distribution with mean and standard deviation |

InverseGaussianDistribution[,] | inverse Gaussian distribution with mean and scale |

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.

ChiSquareDistribution[] | distribution with degrees of freedom |

InverseChiSquareDistribution[] | inverse distribution with degrees of freedom |

FRatioDistribution[n,m] | -ratio distribution with n numerator and m denominator degrees of freedom |

StudentTDistribution[] | Student t distribution with degrees of freedom |

NoncentralChiSquareDistribution[,] | noncentral distribution with degrees of freedom and noncentrality parameter |

NoncentralStudentTDistribution[,] | noncentral Student t distribution with degrees of freedom and noncentrality parameter |

NoncentralFRatioDistribution[n,m,] | noncentral -ratio distribution with n numerator degrees of freedom and m denominator degrees of freedom and numerator noncentrality parameter |

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[n, m, ] 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.

TriangularDistribution[{a,b}] | symmetric triangular distribution on the interval |

TriangularDistribution[{a,b},c] | triangular distribution on the interval with maximum at c |

UniformDistribution[{min,max}] | uniform distribution on the interval |

Piecewise linear distributions.

The triangular distribution TriangularDistribution[{a, b}, c] is a triangular distribution for with maximum probability at and . If is , TriangularDistribution[{a, b}, c] is the symmetric triangular distribution TriangularDistribution[{a, b}].

The uniform distribution UniformDistribution[{min, max}], 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[a, b], 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[, 1/]. If a random variable follows InverseGammaDistribution[1/2, /2], 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[k, ] 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[dist, x] 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[dist, x] evaluates the density at if is a numerical value, and otherwise leaves the function in symbolic form. Similarly, CDF[dist, x] gives the cumulative distribution.

The inverse CDF InverseCDF[dist, q] gives the value of at which CDF[dist, x] reaches . The median is given by InverseCDF[dist, 1/2]. 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[dist, q] and Quantile[dist, q] 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[dist, t] 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[g[x], xdist] 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.

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