Random Variables and Probability Density Functions

Module by: Don Johnson

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

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 , PrA⋃B=PrA+PrB⇒PX x 2 =PX 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 PX x 2 ≥PX x 1 P X x 2 P X x 1 , x 1 ≤ x 2 x 1 x 2 .

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

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.

Comments, questions, feedback, criticisms?

Send feedback

• E-mail the author of the module, Random Variables and Probability Density Functions

Footer

More about this content: Metadata | Version History

• How to reuse and attribute this content
• How to Cite and attribute this content

This work is licensed by Don Johnson under a Creative Commons Attribution License (CC-BY 1.0).

Last edited by Charlet Reedstrom on Aug 1, 2003 4:19 pm GMT-5.