ProbabilityDistributions
:ProbabilityAxioms -- ProbabilityApplications
:RandomVariable -- CumulativeDistributionFunction -- ProbabilityDensity
A probability distribution describes a special universe, a set of real numbers (see AnaLysis) and how probability is distributed among them to determine a RandomVariable. For every RandomVariable, there is a function called the CumulativeDistributionFunction which provides the probability that a given value is not exceeded by the random variable.
:F(x) = Pr[X<x]
If the RandomVariable is a DiscreteRandomVariable, all of the probability is concentrated on a discrete set of points. We can define the probability for a specific point by
:p(x) = limit F(x+t) - F(x-t) as t goes to zero.
For a ContinuousRandomVariable, we define the ProbabilityDensity by
F(x+t) - F(x)
f(x) = F'(x) = limit ------------- as t goes to zero.
t
Several probability distributions are so important that they have been given specific names, the NormalDistribution, the BinomialDistribution, the PoissonDistribution are just three of them.