We note that, by Theorem 2.29, var(X) = X xIm X (x )2P(X = x), (2.34)

We note that, by Theorem 2.29, var(X) =

X x∈Im X

(x − μ)2P(X = x), (2.34)

where μ = E(X). A rough motivation for this definition is as follows. If the dispersion of X about its expectation is very small, then |X − μ| tends to be small, giving that var(X) = E(|X−μ|2) is small also; on the other hand, if there is often a considerable difference between X and its mean, then |X − μ| may be large, giving that var(X) is large also.

Equation (2.34) is not always the most convenient way to calculate the variance of a discrete random variable.We may expand the term (x − μ)2 in (2.34) to obtain var(X) =

X x

(x2 − 2μx + μ2)P(X = x)

=

X x

x2P(X = x) − 2μ

X x

xP(X = x) + μ2 X

x P(X = x)

= E(X2) − 2μ2 + μ2 by (2.28) and (2.6)

= E(X2) − μ2, where μ = E(X) as before. Thus we obtain the useful formula var(X) = E(X2) − E(X)2. (2.35)