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

5.3 INDICATOR RANDOM VARIABLE

For the abstract probability space , a basic type of random variable is the indicator random variable.

Definition: Indicator Random Variable The indicator random variable is a mapping of event to the real line as follows:

(5.3)

The mapping for the indicator random variable is shown in Figure 5.2. Observe that Ec necessarily maps to zero. The indicator random variable is essentially the same as the indicator function, except that its argument ω is random as specified by the probability space. Its expectation turns out to be the probability of event E:

FIGURE 5.1 Four important moments of random variable X.

FIGURE 5.2 Mapping of the indicator function for event E to the indicator random variable X on the real line .

(5.4)

and also

(5.5)

These probabilities ...