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Random Variables and Probability Density Functions

Module by: Don Johnson. E-mail the author

A random variable XX is the assignment of a number - real or complex - to each sample point in sample space. Thus, a random variable can be considered a function whose range is a set and whose ranges are, most commonly, a subset of the real line. The probability distribution function or cumulative is defined to be

P X x=PrXx P X x X x
(1)
Note that XX denotes the random variable and xx denotes the argument of the distribution function. Probability distribution functions are increasing functions: if A=ω Xω x 1 A ω X ω x 1 and B=ω x 1 <Xω x 2 B ω x 1 X ω x 2 , (PrAB=PrA+PrB)(P X x 2 =P X x 1 +Pr x 1 <X x 2 ) A B A B P X x 2 P X x 1 x 1 X x 2 , which means that P X x 2 P X x 1 P X x 2 P X x 1 , x 1 x 2 x 1 x 2 .

The probability density function p X x p X x is defined to be that function when integrated yields the distribution function.

P X x=xp X αdα P X x α x p X α
(2)
As distribution functions may be discontinuous, we allow density functions to contain impulses. Furthermore, density functions must be non-negative since their integrals are increasing.

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