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

5.17 EXPECTATION AND THE INDICATOR FUNCTION

Sometimes it is convenient to write various probabilistic quantities in terms of an expectation using the indicator function. For random variable X with pdf FX(x) and cdf FX(x), observe that

(5.222)

where in this context, I[a,b](X) is a particular nonlinear transformation of random variable X. Likewise, we can write the cdf as follows:

(5.223)

We might also be interested in various expectations of the indicator function itself. For example, observe that

(5.224)

where (5.222) has been substituted for the mean of the indicator function. Since squaring I[a,b](X) gives the same indicator function, we finally have

(5.225)

As with any function of random variable X, this variance can be computed without first finding the pdf of the mapping. A special case of the result above is

(5.226)

For random variables X and Y, the following covariance can be computed using a set operation:

(5.227)

Since the indicator function takes on only two values ...