POL 571: Expectation and Functions of Random Variables

where F (x) is the distribution function of X. The expectation operator has inherits its properties from those of summation and integral. In particular, the following theorem shows that expectation preserves the inequality and is a linear operator. Theorem 1 (Expectation) Let X and Y be random variables with finite expectations. 1. If g(x) ≥ h(x) for all x ∈ R, then E[g(X)] ≥ E[h(X)]. 2. E(aX + bY + c) = aE(X) + bE(Y ) + c for any a, b, c ∈ R.