if the last sum converges absolutely. 2 Two simple but useful properties of expectation are as follows.

Theorem 2.30 Let X be a discrete random variable and let

a, b ∈ R.

(a) If P(X ≥ 0) = 1 and E(X) = 0, then P(X = 0) = 1.

(b) We have that E(aX +

b) = aE(X) + b.

Proof

(a) Suppose the assumptions hold. By the definition (2.28) of E(X), we have that xP(X = x) = 0 for all x ∈ Im X. Therefore, P(X = x) = 0 for x 6= 0, and the claim follows.

(b) This is a simple consequence of Theorem 2.29 with g(x) = ax +

b. 2