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# Continuous Distributions

The functions described here are among the most commonly used continuous statistical distributions. You can compute their densities, means, variances, and other related properties. The distributions themselves are represented in the symbolic form name[param1, param2, ...]. Functions such as Mean, which give properties of statistical distributions, take the symbolic representation of the distribution as an argument. "Discrete Distributions" describes many discrete 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 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 is proportional to the distribution NormalDistribution[0, 1/( Sqrt[2/])] limited to the domain [0, ).
The inverse Gaussian distribution , sometimes called the Wald distribution, is the distribution of first passage times in Brownian motion with positive drift.
 ChiSquareDistribution[] 2 distribution with degrees of freedom FRatioDistribution[n,m] F-ratio distribution with n numerator and m denominator degrees of freedom StudentTDistribution[] Student t distribution with degrees of freedom NoncentralChiSquareDistribution[,] noncentral 2 distribution with degrees of freedom and noncentrality parameter NoncentralStudentTDistribution[,] noncentral Student t distribution with degrees of freedom and noncentrality parameter NoncentralFRatioDistribution[n,m,] noncentral F-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 X1,..., X are independent normal random variables with unit variance and mean zero, then has a 2 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 2 distribution is most typically used when describing the variance of normal samples.
A variable that has a Student t distribution can also be written as a function of normal random variables. Let X and Z be independent random variables, where X is a standard normal distribution and Z is a 2 variable with degrees of freedom. In this case, has a t distribution with degrees of freedom. The Student t distribution is symmetric about the vertical axis, and characterizes the ratio of a normal variable to its standard deviation. When =1, the t distribution is the same as the Cauchy distribution.
The F-ratio distribution is the distribution of the ratio of two independent 2 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 2=1 and nonzero means follows a noncentral 2 distribution . 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, /2 or is used as the noncentrality parameter.
The noncentral Student t distribution describes the ratio where is a central 2 random variable with degrees of freedom, and X is an independent normally distributed random variable with variance 2=1 and mean .
The noncentral F-ratio distribution is the distribution of the ratio of to , where is a noncentral 2 random variable with noncentrality parameter and n1 degrees of freedom and is a central 2 random variable with m degrees of freedom.
 TriangularDistribution[{a,b}] symmetric triangular distribution on the interval {a, b} TriangularDistribution[{a,b},c] triangular distribution on the interval {a, b} with maximum at c UniformDistribution[{min,max}] uniform distribution on the interval {min, max}

Piecewise linear distributions.

The triangular distribution TriangularDistribution[{a, b}, c] is a triangular distribution for a<X<b with maximum probability at c and a<c<b. If c 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 LaplaceDistribution[,] Laplace (double exponential) distribution with mean and scale 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 X is uniformly distributed on[-, ], then the random variable tan (X) follows a Cauchy distribution CauchyDistribution[a, b], with a=0 and b=1.
When =n/2 and =2, the gamma distribution describes the distribution of a sum of squares of n-unit normal random variables. This form of the gamma distribution is called a 2 distribution with degrees of freedom. When =1, the gamma distribution takes on the form of the exponential distribution , often used in describing the waiting time between events.
When X1 and X2 have independent gamma distributions with equal scale parameters, the random variable follows the beta distribution , where and are the shape parameters of the gamma variables.
The distribution is followed by the square root of a 2 random variable. For n=1, the distribution is identical to with . For n=2, the distribution is identical to the Rayleigh distribution with =1. For n=3, the distribution is identical to the Maxwell-Boltzmann distribution with =1.
The Laplace distribution is the distribution of the difference of two independent random variables with identical exponential distributions. The logistic distribution is frequently used in place of the normal distribution when a distribution with longer tails is desired.
The Pareto distribution may be used to describe income, with k representing the minimum income possible.
The Weibull distribution is commonly used in engineering to describe the lifetime of an object. The extreme value distribution 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, . 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] qth 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 (t) ExpectedValue[f,dist] expected value of the pure function f in dist ExpectedValue[f[x],dist,x] expected value of f[x] for x in dist RandomReal[dist] pseudorandom number with specified distribution RandomReal[dist,dims] pseudorandom array with dimensionality dims, and elements from the specified distribution

Functions of statistical distributions.

The cumulative distribution function (cdf) at x is given by the integral of the probability density function (pdf) up to x. 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 x if x is a numerical value, and otherwise leaves the function in symbolic form. Similarly, CDF[dist, x] gives the cumulative distribution. Domain[dist] gives the domain of PDF[dist, x] and CDF[dist, x].
The inverse cdf InverseCDF[dist, q] gives the value of x at which CDF[dist, x] reaches q. The median is given by InverseCDF[dist, 1/2]. Quartiles, deciles and percentiles are particular values of the inverse cdf. 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 yx f (x)x, where f (x) is the pdf of the distribution. The variance Variance[dist] is given by (x-)2f (x)x. 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 (t)=f (x)exp (itx)x. In the discrete case, (t)=f (x)exp (itx). Each distribution has a unique characteristic function, which is sometimes used instead of the pdf to define a distribution.
The expected value ExpectedValue[g, dist] of a function g is given by f (x)g (x)x. In the discrete case, the expected value of g is given by f (x)g (x). ExpectedValue[g[x], dist, x] is equivalent to ExpectedValue[g, dist].
RandomReal[dist] gives pseudorandom numbers from the specified distribution.
This gives a symbolic representation of the gamma distribution with =3 and =1.
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Here is the cumulative distribution function evaluated at 10.
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This is the cumulative distribution function. It is given in terms of the built-in function GammaRegularized.
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Here is a plot of the cumulative distribution function.
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This is a pseudorandom array with elements distributed according to the gamma distribution.
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