Home page for accesible maths 5.6 Poisson random variables 5.6 Poisson random variables 5.7 Negative Binomial random variables (not examinable)

Style control - access keys in brackets

Font (2 3) - + Letter spacing (4 5) - + Word spacing (6 7) - + Line spacing (8 9) - +

5.6.1 Poisson approximation to the Binomial

One use of Poisson random variables for rare event modelling. Consider a Binomial random variable ${R}$ from ${n}$ trials with the probability of success being ${\theta}$ . A rare event has a small probability of occurring, but when there are a large number of trials, the probability that one or more events occur is not negligible.

Define ${\lambda=n\theta}$ . We will send $n\to\infty$ while keeping $\lambda$ fixed (and so $\theta=\lambda/n$ ).

To get our result, we will need the following result from calculus:

\displaystyle\lim_{n{\rightarrow}\infty}\left(1+\frac{x}{n}\right)^{n}=e^{x}% \quad[=\exp(x)]

for all $x\in{\mathbb{R}}$ .

Now we calculate

$\displaystyle{\rm P}(R=r)$	$\displaystyle=$	$\displaystyle\binom{n}{r}\theta^{r}(1-\theta)^{n-r}$
	$\displaystyle=$	$\displaystyle\frac{n!}{r!(n-r)!}\left(\frac{\lambda}{n}\right)^{r}\left(1-% \frac{\lambda}{n}\right)^{n-r}$
	$\displaystyle=$	$\displaystyle\frac{\lambda^{r}}{r!}\underbrace{\left(1-\frac{\lambda}{n}\right% )^{n}}_{{\rightarrow}\exp(-\lambda)}\underbrace{\left(1-\frac{\lambda}{n}% \right)^{-r}}_{{\rightarrow}1}\underbrace{\frac{n!}{(n-r)!n^{r}}}_{{% \rightarrow}1}$
	$\displaystyle{\rightarrow}$	$\displaystyle\frac{\lambda^{r}}{r!}\exp(-\lambda),$

where the limits are taken as $n\to\infty$ .

Rare events: The binomial pmf with large ${n}$ and small ${\theta}$ can be approximated by a Poisson pmf with parameter ${\lambda=n\theta}$ . For a good approximation ${n\geq 100}$ and ${\theta\leq.01}$ .

Example 5.24.

An outbreak of poliomyelitis (Guillain-Barre syndrome =GBS) occurred in Finland in 1984 and as a consequence an intensive vaccination programme occurred. Before the vaccination programme the average number of cases of GBS was ${3.67}$ per month. In the month after the vaccination programme there were ${14}$ cases of GBS reported.

How unusual is this? Compute the p-value = ${{\rm P}}$ (observed value or worse), under the assumption of natural Poisson variability. It is ${{\rm P}(R\geq 14)}$ and measures the worry in observing ${14}$ cases.

R hint: 1-sum( dpois(0:13,lambda=3.67) )

Solution.

We could count the number of susceptible people, and estimate the probability that each catches GBS. However, since this is a rare event, we can instead model the number of cases each month as ${R\sim{\rm Pois}(3.67)}$ :

\displaystyle p_{R}(r)=\frac{3.67^{r}\exp(-3.67)}{r!}

for $r=0,1,2,,\dots$ . The p-value is

\displaystyle{\rm P}(R\geq 14)=1-\sum_{r=0}^{13}p(r)=0.000031.

Very small and so worrying: the vaccination appears to have caused excess cases. Worry!