A random variable—unlike a normal variable—does not have a specific value, but rather a range of values and a density that gives different probabilities of obtaining values for each subset. This can be used to model uncertainty, whether from incomplete or simplified models. Random variables are used extensively in areas such as social science, science, engineering, and finance. The Wolfram Language uses symbolic distributions to represent a random variable. In the Wolfram Language, you can directly compute several dozen properties from symbolic distributions, including finding the probability of an arbitrary event or simulating it to generate data. The Wolfram Language has the largest collection of parametric distributions ever assembled, and parametric distributions can be automatically estimated from data. The Wolfram Language provides nonparametric distributions directly computed from data, automating and generalizing the many nonparametric methods in use for specific properties. Distributions can be derived from other distributions or given by formulas for distribution functions, giving infinite extensibility to the whole framework.

Probability — compute probabilities of predicates given distributions

Expectation — compute expectations of expressions given distributions

NProbability  ▪  NExpectation  ▪  Distributed ()  ▪  Conditioned ()

Simulation & Estimation

RandomVariate — generate random variates from a distribution

EstimatedDistribution — estimate parametric or derived distribution from data

FindDistributionParameters — find parameter estimates for a particular distribution

FindDistribution — try to find a distribution with a simple functional form to fit data

Hypothesis Testing »

DistributionFitTest — test how well data and a distribution fit

LocationTest  ▪  VarianceTest  ▪  LocationEquivalenceTest  ▪  ...

AndersonDarlingTest  ▪  KolmogorovSmirnovTest  ▪  MannWhitneyTest  ▪  ...

Distribution-Related Functions »

PDF — probability density function

CDF — cumulative distribution function

SurvivalFunction  ▪  HazardFunction  ▪  InverseCDF  ▪  Quantile  ▪  RarerProbability  ▪  ...

Moments and Generating Functions »

Moment — moments of distributions and data

Cumulant  ▪  MomentGeneratingFunction  ▪  MomentConvert  ▪  ...

Parametric Distributions »

NormalDistribution — univariate normal distribution

MultinormalDistribution — multivariate normal distribution

StableDistribution  ▪  MaxStableDistribution  ▪  PoissonDistribution  ▪  ...

Nonparametric Distributions »

HistogramDistribution — distribution constructed from a histogram of data

SmoothKernelDistribution — distribution constructed from smoothing of data

EmpiricalDistribution  ▪  SurvivalDistribution  ▪  KernelMixtureDistribution

Derived Distributions »

TransformedDistribution — distribution of a function of a random variable

CopulaDistribution — distribution from kernel and marginal distributions

TruncatedDistribution  ▪  CensoredDistribution  ▪  OrderDistribution  ▪  ...

Formula Distributions

ProbabilityDistribution — distribution constructed from a distribution function

Categorical Distributions

CategoricalDistribution — distribution for finite categories of data

Matrix Distributions »

WishartMatrixDistribution — matrix-valued chi-squared distribution

GaussianOrthogonalMatrixDistribution — symmetric matrices (GOE)

CircularRealMatrixDistribution  ▪  MatrixPropertyDistribution  ▪  ...

Statistical Visualization »

QuantilePlot — quantile-quantile plot of distributions and data

ProbabilityScalePlot — normal plot, Weibull plot, etc.

ProbabilityPlot  ▪  Histogram  ▪  SmoothHistogram  ▪  DensityHistogram  ▪  ...

BoxWhiskerChart  ▪  DistributionChart

Computable Data

Import — import data from a variety of formats

ExampleData — special statistics data collection

Country City Chemical Company Person Planet Food Species Building University ...