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

5.10 EXPECTATION OF A FUNCTION

The expectation of function Y = g(X) of random variable X can be computed in two ways: (i) deriving the pdf fY(y) and then calculating or (ii) directly calculating based on FX(x). We demonstrate the equivalence of both approaches in the following theorem.

Theorem 5.4 The expectation of function Y = g(X) of random variable X is

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Proof. (Discrete random variable). The proof is a straightforward application of the mapping process from X to Y:

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The inner sum accounts for all values of x (and the corresponding probability mass) that map to value y. The double sum is the same as a single sum over all possible values of g(x), weighted by pX[x]:

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(Continuous random variable). Assume that g(X) is monotonically increasing where each x maps to a unique y. Thus

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Substituting y = g(x) and x = g−1(y) gives

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where dx/dy>0 for a monotonically ...