5.2. Convergence in Distribution 337Example 5.2.6.LetYnhave a distribution that isb(n, p). Suppose that the mean
μ=npis the same for everyn;thatis,p=μ/n,whereμis a constant. We shall
find the limiting distribution of the binomial distribution, whenp=μ/n, by finding
the limit ofMYn(t). Now
MYn(t)=E(etYn)=[(1−p)+pet]n=[
1+
μ(et−1)
n]nfor all real values oft. Hence we have
lim
n→∞
MYn(t)=eμ(et−1)for all real values oft. Since there exists a distribution, namely the Poisson distribu-
tion with meanμ,thathasmgfeμ(e
t−1)
, then, in accordance with the theorem and
under the conditions stated, it is seen thatYnhas a limiting Poisson distribution
with meanμ.
Whenever a random variable has a limiting distribution, we may, if we wish, use
the limiting distribution as an approximation to the exact distribution function. The
result of this example enables us to use the Poisson distribution as an approximation
to the binomial distribution whennis large andpis small. To illustrate the use
of the approximation, letY have a binomial distribution withn=50andp= 251.
Then, using R for the calculations, we have
Pr(Y≤1) = (^2425 )^50 + 50( 251 ) = pbinom(1,50,1/25) = 0. 4004812
approximately. Sinceμ=np= 2, the Poisson approximation to this probability is
e−^2 +2e−^2 = ppois(1,2) = 0. 4060058.
Example 5.2.7.LetZnbeχ^2 (n). Then the mgf ofZnis (1− 2 t)−n/^2 ,t<^12 .The
mean and the variance ofZnare, respectively,nand 2n. The limiting distribution
of the random variableYn=(Zn−n)/
√
2 nwill be investigated. Now the mgf of
YnisMYn(t)=E{
exp[
t(
Zn−n
√
2 n)]}= e−tn/√
2 nE(etZn/
√
2 n)=exp[
−(
t√
2
n)
(n
2)
](
1 − 2t
√
2 n)−n/ 2
,t<√
2 n
2.This may be written in the formMYn(t)=(
et√
2 /n−t√
2
net√
2 /n)−n/ 2
,t<√
n
2.In accordance with Taylor’s formula, there exists a numberξ(n), between 0 and
t
√
2 /n, such thatet√
2 /n=1+t√
2
n
+1
2(
t√
2
n) 2
+eξ(n)
6(
t√
2
n) 3
.