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Probability & Statistics
Probability and statistics are used to model uncertainty from a variety of sources, such as incomplete or simplified models. Yet you can build useful models for aggregate or overall behavior of the system in question. Such a model is that of a probability distribution.
Mathematica uses symbolic distributions. 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
Simulation & Estimation
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
Formula 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
WeatherData  ▪ FinancialData  ▪ CountryData  ▪ WordData  ▪ CityData  ▪ ...