2.5 Conditional expectation and the partition theorem 33 Example 2.36 If X has the geometric distribution with parameter p (= 1 − q), the mean of X is E(X) =

∞X k=1 kpqk−1

=

p

(1 − q)2 =

1 p

, and the variance of X is var(X) =

∞X k=1 k2 pqk−1 −

1 p2 by (2.35).4 Now,

∞X k=1 k2qk−1 = q

∞X k=1 k(k − 1)qk−2 +

∞X k=1 kqk−1

=

2q

(1 − q)3 +

1

(1 − q)2 by Footnote 4, giving that var(X) = p



2q p3 +

1 p2



−

1 p2

= qp−2. △