5.3The Hypergeometric Random Variable 159
throughout the region, a new catch of size, say,nis made. LetXdenote the number of
marked animals in this second capture. If we assume that the population of animals in the
region remained fixed between the time of the two catches and that each time an animal
was caught it was equally likely to be any of the remaining uncaught animals, it follows
thatXis a hypergeometric random variable such that
P{X=i}=(r
i)(N−r
n−i)(
N
n) ≡Pi(N)Suppose now thatXis observed to equali. That is, the fractioni/nof the animals in
the second catch were marked. By taking this as an approximation ofr/N, the proportion
of animals in the region that are marked, we obtain the estimatern/iof the number of
animals in the region. For instance, ifr=50 animals are initially caught, marked, and
then released, and a subsequent catch ofn=100 animals revealedX =25 of them
that were marked, then we would estimate the number of animals in the region to be
about 200. ■
There is a relationship between binomial random variables and the hypergeo-
metric distribution that will be useful to us in developing a statistical test concerning
two binomial populations.
EXAMPLE 5.3c LetXandYbe independent binomial random variables having respective
parameters (n,p) and (m,p). The conditional probability mass function ofXgiven that
X+Y=kis as follows.
P{X=i|X+Y=k}=P{X=i,X+Y=k}
P{X+Y=k}=P{X=i,Y=k−i}
P{X+Y=k}=P{X=i}P{Y=k−i}
P{X+Y=k}=(
n
i)
pi(1−p)n−i(
m
k−i)
pk−i(1−p)m−(k−i)
(
n+m
k)
pk(1−p)n+m−k=(
n
i)(
m
k−i)(
n+m
k)