Let Y be a discrete random variable with probability function p(y) and mean
E(Y ) = μ; then
V(Y ) = σ2 = E[(Y − μ)2] = E(Y 2) − μ2.
Proof
σ2 = E[(Y − μ)2] = E(Y 2 − 2μY + μ2)
= E(Y 2) − E(2μY ) + E(μ2) (by Theorem 3.5).
Noting that μ is a constant and applying Theorems 3.4 and 3.3 to the second
and third terms, respectively, we have
σ2 = E(Y 2) − 2μE(Y ) + μ2.
But μ = E(Y ) and, therefore,
σ2 = E(Y 2) − 2μ2 + μ2 = E(Y 2) − μ2.

是不是所有情况下假设E(Y ) = μ都成立?