Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

5.4 SIMPLE RANDOM VARIABLE

A simple random variable is a generalization of the indicator random variable where instead of two events, N mutually exclusive events in that form a partition of Ω are mapped to N values in .

Definition: Simple Random Variable Simple random variable X has the form

(5.9)

where xn is the value in assigned to event En, and the {En} form a partition of Ω.

A simple random variable is essentially the same as a simple function (see Appendix D), except that its argument ω is random as determined by the probability space . It follows from this definition that

(5.10)

Clearly, any discrete random variable with a finite number of outcomes is a simple random variable because it is readily represented by (5.9). The expectation is

(5.11)

where (5.10) has been used, and ...