Advanced High-School Mathematics

330 CHAPTER 6 Inferential Statistics

Next, ifX is the binomial random variable with success probabilityp,
then we may write

X = B 1 +B 2 +···+Bn,

where eachBi is a Bernoulli random variable. It follows easily from
what we already proved above that

E(X) = E(B 1 ) +E(B 2 ) +···E(Bn) = np,

and

Var(X) = Var(B 1 ) + Var(B 2 ) +···+ Var(Bn) = np(1−p).

6.1.6 Generalizations of the geometric distribution

Generalization 1: The negative binomial distribution

Suppose that we are going to perform a numberX of Bernoulli trials,
each with success probabilityp, stopping after exactlyrsuccesses have
occurred. Then it is clear that

P(X=x) =

Ñ

x− 1

r− 1

é

pr(1−p)x−r.

In order to compute the mean and variance ofXnote thatXis easily
seen to be the sum of geometric random variablesG 1 , G 2 , ...,Gr, where
the success probability of each isp:

X = G 1 +G 2 +···+Gr.

Using the results of (6.1.4) we have, for eachi= 1, 2 , ...,r, that

E(Gi) =

1

p

, Var(Gi) =

1 −p

p^2

.

It follows, therefore, that the mean and variance of the negative bino-
mial random variableX are given by