Advanced High-School Mathematics

SECTION 6.1 Discrete Random Variables 337

Var(X) = E(X^2 )−E(X)^2

=

n(n−1)k(k−1)

N(N−1)

+

nk

N

−

(nk

N

) 2

=

nk(N−n)(N−k)

N^2 (N−1)

6.1.8 The Poisson distribution

The Poisson random variable can be thought of as the limit of a
binomial random variable in the following sense. First of all, assume
thatY is the binomial random variable which measures the number of
successes in n trials and where the probability of each trial is p. As
we saw above, the mean of this random variable isμY = np. Now,
rather than limiting the number of trials, we take the limit asn→∞
but holding fixed the meanμ=μY. We call the resulting random
variable thePoisson random variable with meanμ. If we denote
this byX, then the distribution ofXis computed as follows:

P(X=k) = limn→∞

Ñ

n

k

é

pk(1−p)n−k

= limn→∞

Ñ

n

k

éÇ

μ

n

åkÇ

1 −

μ

n

ån−k

(sinceμ=np)

= limn→∞

n(n−1)···(n−k+ 1)

k!

Çμ

n

åkÇ

1 −

μ

n

ån−k

= limn→∞

n(n−1)···(n−k+ 1)

nk

Ñ

μk

k!

éÇ

1 −

μ

n

ån−k

=

Ñ

μk

k!

é

nlim→∞

Ç

1 −

μ

n

ån−k Ç

since limn→∞

n(n−1)···(n−k+ 1)

nk = 1

å

=

Ñ

μk

k!

é

nlim→∞

Ç

1 −

μ

n

ånÇ

1 −

μ

n

å−k

=

Ñ

μk

k!

é

nlim→∞

Ç

1 −

μ

n

ån