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

5.9 PROPERTIES OF EXPECTATION

The following properties of expectation apply to discrete, continuous, and mixed random variables:

• Indicator function. The expectation of the indicator function is a probability: (5.56) This is easily seen as follows: (5.57) where FX(x) is the cdf of FX(x).
• Linearity. Expectation is a linear operator: (5.58) where {a, b} are constant coefficients.
• Nonnegative. If X is nonnegative such that P(X<0) = 0, then because .
• Symmetry. If there exists a value μ about which the pdf is symmetric, which is the case for a Gaussian random variable, then μ is the expected value (the mean). However, this is not true for some distributions such as the Cauchy random variable mentioned earlier, where the integral is not finite when evaluated at each limit.
• Independence. If are independent random variables, then (5.59) This follows because the joint pdf splits into the product ...