[機率] Summing a Random Number of Random Variables

Reference: http://www.tecknowbasic.com/EE126/ee126-lec14.pdf

Suppose Y = X_1 + ... + X_N, the X_i are iid, but N is a random variable independent of the X_i's.  Then 

• E[Y] = E[N]E[X]
• Var(Y) = E[X_i]^2 Var(N) + E[N] Var(X_i)