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

5.7 SUMMARY OF EXPECTATION

In this section, we summarize the main results for the expectation of a random variable. For the abstract probability space , the expectation of random variable X is

(5.51)

where is the probability measure. In the probability space , the expectation of random variable X is the following integral:

(5.52)

where x is the integrand and the cdf FX(x) is the integrator. If FX(x) is differentiable everywhere, then this last expression can be expressed as an ordinary Riemann integral by substituting dFX(x) = fX(x)dx:

(5.53)

where FX(x) is the pdf (and xfX(x) is the integrand). This result can be used for a discrete random variable (as well as a mixed random variable) if Dirac delta functions are used to represent the pdf. For a discrete random variable with pmf pX[x], the expectation is also evaluated as follows:

(5.54)

where the sifting property of the Dirac delta ...