3.6.1 The Poisson Distribution as a Limit

The rationale for using the Poisson distribution in many situations is provided by the following proposition.

Proposition

Suppose that in the binomial pmf $b\left(x;n,p\right)$ , we let $n\to \infty$ and $p\to 0$ in such a way that $np\to \mu >0.$ Then $b\left(x;n,p\right)\to p\left(x;\mu \right).$

According to this result, in any binomial experiment in which $n$ is large and $p$ is small, $b\left(x;n,p\right)\approx p\left(x;\mu \right)$ , where $\mu =np$. As a rule of thumb, this approximation can safely be applied if $n>50$ and $np<5$.

EX 3.39 error page

Table 3.2 shows the Poisson distribution for $\mu =3$ along with three binomial distributions with $np=3$.

Table 3.2 Comparing the Poisson and Three Binomial Distributions

$\mathbf{x}$	$\mathbf{n}=30,\mathbf{p}=.1$	$\mathbf{n}=100,\mathbf{p}=.03$	$\mathbf{n}=300,\mathbf{p}=.01$	Poisson, $\mathbf{\mu }=3$
0	0.042391	0.047553	0.049041	0.049787
1	0.141304	0.147070	0.148609	0.149361
2	0.227656	0.225153	0.224414	0.224042
3	0.236088	0.227474	0.225170	0.224042
4	0.177066	0.170606	0.168877	0.168031
5	0.102305	0.101308	0.100985	0.100819
6	0.047363	0.049610	0.050153	0.050409
7	0.018043	0.020604	0.021277	0.021604
8	0.005764	0.007408	0.007871	0.008102
9	0.001565	0.002342	0.002580	0.002701
10	0.000365	0.000659	0.000758	0.000810

Figure 3.8 plots the Poisson along with the first two binomial distributions. The approximation is of limited use for $n=30$ , but of course the accuracy is better for $n=100$ and much better for $n=300$ .

Figure 3.8 Comparing a Poisson and two binomial distributions