Probability_Distributions
back to Probability and Statistics
:Probability Axioms -- Probability Applications
:Random Variable -- Cumulative Distribution Function -- Probability Density
A Probability Distribution describes a special universe, a set of real numbers (see AnaLysis) and how probability is distributed among them to determine a Random Variable. For every Random Variable, there is a function called the Cumulative Distribution Function which provides the probability that a given value is not exceeded by the random variable.
:F(x) = Pr[X<x]
If the Random Variable is a Discrete Random Variable, 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 Continuous Random Variable, we define the Probability Density (at x) 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 Normal Distribution, the Binomial Distribution], the [[Poisson Distribution are just three of them.