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 the Probability Mass Function
:p(x) = limit F(x+t) - F(x-t) as t goes to zero.
For a Continuous Random Variable, we define the Probability Density Function (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.----
For Continous Random Variable you ARE giving the Cumulative Distribution Function, not the Probability Density Function...eh? The probability that a continuous random variable X
takes a value less than or equal to x is denoted Pr(X<=x). The probability density function of X, where X is a continuous random variable, is the function f such that
- Pr(a<=X<=b)= INTEGRAL ( as x ranges from a thru b) f(x) dx.