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.
Mathematica uses symbolic distributions to represent a random variable. In Mathematica, you can directly compute several dozen properties from symbolic distributions, including finding the probability of an arbitrary event or simulating it to generate data. Mathematica has the largest collection of parametric distributions ever assembled, and parametric distributions can be automatically estimated from data. Mathematica 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
RandomVariate — generate random variates from a distribution
EstimatedDistribution — estimate parametric or derived distribution from data
FindDistributionParameters — find parameter estimates as rules
DistributionFitTest — test how well data and a distribution fit
PDF — probability density function
CDF — cumulative distribution function
Moment — moments of distributions and data
NormalDistribution — univariate normal distribution
MultinormalDistribution — multivariate normal distribution
HistogramDistribution — distribution constructed from a histogram of data
SmoothKernelDistribution — distribution constructed from smoothing of data
TransformedDistribution — distribution of a function of a random variable
CopulaDistribution — distribution from kernel and marginal distributions
ProbabilityDistribution — distribution constructed from a distribution function
QuantilePlot — quantile-quantile plot of distributions and data
ProbabilityScalePlot — normal plot, Weibull plot, etc.
Import — import data from a variety of formats
ExampleData — special statistics data collection