Introduction to Probability and Statistics for Engineers and Scientists

158 Chapter 5: Special Random Variables

Also, fori<j,

Cov(Xi,Xj)=E[XiXj]−E[Xi]E[Xj]

Now, because bothXiandXjare Bernoulli (that is, 0−1) random variables, it follows
thatXiXjis a Bernoulli random variable, and so

E[XiXj]=P{XiXj= 1 }

=P{Xi=1,Xj= 1 }

=

N(N−1)

(N+M)(N+M−1)

from Equation 5.3.3 (5.3.7)

So from Equation 5.3.2 and the foregoing we see that fori=j,

Cov(Xi,Xj)=

N(N−1)

(N+M)(N+M−1)

−

(

N

N+M

) 2

=

−NM

(N+M)^2 (N+M−1)

Hence, since there are

(n

2

)
terms in the second sum on the right side of Equation 5.3.5,
we obtain from Equation 5.3.6

Var(X)=

nNM

(N+M)^2

−

n(n−1)NM

(N+M)^2 (N+M−1)

=

nNM

(N+M)^2

(

1 −

n− 1

N+M− 1

)

(5.3.8)

If we letp=N/(N+M) denote the proportion of batteries in the bin that are acceptable,
we can rewrite Equations 5.3.4 and 5.3.8 as follows.

E(X)=np

Var(X)=np(1−p)

[

1 −

n− 1

N+M− 1

]

It should be noted that, for fixedp,asN+M increases to∞, Var(X) converges to
np(1−p), which is the variance of a binomial random variable with parameters (n,p).
(Why was this to be expected?)

EXAMPLE 5.3b An unknown number, say N, of animals inhabit a certain region.
To obtain some information about the population size, ecologists often perform the
following experiment: They first catch a number, sayr, of these animals, mark them
in some manner, and release them. After allowing the marked animals time to disperse