Random Variables

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.

(RefLink[Probability, paclet:ref/Probability]) — compute probabilities of predicates given distributions

(RefLink[Expectation, paclet:ref/Expectation]) — compute expectations of expressions given distributions

Simulation & Estimation

(RefLink[RandomVariate, paclet:ref/RandomVariate]) — generate random variates from a distribution

(RefLink[EstimatedDistribution, paclet:ref/EstimatedDistribution]) — estimate parametric or derived distribution from data

(RefLink[FindDistributionParameters, paclet:ref/FindDistributionParameters]) — find parameter estimates for a particular distribution

(RefLink[FindDistribution, paclet:ref/FindDistribution]) — try to find a distribution with a simple functional form to fit data

(OrangeLink[Hypothesis Testing », paclet:guide/HypothesisTests])

(RefLink[DistributionFitTest, paclet:ref/DistributionFitTest]) — test how well data and a distribution fit

(OrangeLink[Distribution-Related Functions », paclet:guide/StatisticalDistributionFunctions])

(RefLink[PDF, paclet:ref/PDF]) — probability density function

(RefLink[CDF, paclet:ref/CDF]) — cumulative distribution function

(OrangeLink[Moments and Generating Functions », paclet:guide/StatisticalMomentsAndGeneratingFunctions])

(RefLink[Moment, paclet:ref/Moment]) — moments of distributions and data

(OrangeLink[Parametric Distributions », paclet:guide/ParametricStatisticalDistributions])

(RefLink[NormalDistribution, paclet:ref/NormalDistribution]) — univariate normal distribution

(RefLink[MultinormalDistribution, paclet:ref/MultinormalDistribution]) — multivariate normal distribution

(OrangeLink[Nonparametric Distributions », paclet:guide/NonparametricStatisticalDistributions])

(RefLink[HistogramDistribution, paclet:ref/HistogramDistribution]) — distribution constructed from a histogram of data

(RefLink[SmoothKernelDistribution, paclet:ref/SmoothKernelDistribution]) — distribution constructed from smoothing of data

(OrangeLink[Derived Distributions », paclet:guide/DerivedDistributions])

(RefLink[TransformedDistribution, paclet:ref/TransformedDistribution]) — distribution of a function of a random variable

(RefLink[CopulaDistribution, paclet:ref/CopulaDistribution]) — distribution from kernel and marginal distributions

Formula Distributions

(RefLink[ProbabilityDistribution, paclet:ref/ProbabilityDistribution]) — distribution constructed from a distribution function

(OrangeLink[Matrix Distributions », paclet:guide/MatrixDistributions])

(RefLink[WishartMatrixDistribution, paclet:ref/WishartMatrixDistribution]) — matrix-valued chi-squared distribution

(RefLink[GaussianOrthogonalMatrixDistribution, paclet:ref/GaussianOrthogonalMatrixDistribution]) — symmetric matrices (GOE)

(OrangeLink[Statistical Visualization », paclet:guide/StatisticalVisualization])

(RefLink[QuantilePlot, paclet:ref/QuantilePlot]) — quantile-quantile plot of distributions and data

(RefLink[ProbabilityScalePlot, paclet:ref/ProbabilityScalePlot]) — normal plot, Weibull plot, etc.

Computable Data

(RefLink[Import, paclet:ref/Import]) — import data from a variety of formats

(RefLink[ExampleData, paclet:ref/ExampleData]) — special statistics data collection