Random Variables and Probability Density Functions 1.2 2003/05/13 2003/08/01 16:09:37.007 GMT-5 Don Johnson dhj@rice.edu Don Johnson dhj@rice.edu Eileen Krause erkrause@rice.edu Kyle Clarkson kclarks@rice.edu Elizabeth Gregory lizzardg@rice.edu Kevin Duh kevinduh@rice.edu Mariyah Poonawala mariyah@rice.edu Matthew Jeanes mjeanes@rice.edu Jeffrey Silverman jsilv@rice.edu A random variable X 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 X x Note that X denotes the random variable and x denotes the argument of the distribution function. Probability distribution functions are increasing functions: if A ω X ω x 1 and B ω x 1 X ω 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 , x 1 x 2 . The probability density function p X x is defined to be that function when integrated yields the distribution function. P X x α x p X α 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.