##

The Poisson(*l**<=
span>t*) distribution describes the possible number of eve=
nts that may occur in an exposure of *t* units, where the average nu=
mber of events per unit of exposure is *l*. A Po=
isson(*l**t*) distribution is=
thus the sum of *t* independent Poisson(*l) distributions. We might intuitively guess then that if **l**t* is sufficiently large, a P=
oisson(*l**t*) dis=
tribution will start to look like a Normal distribution, beca=
use of Central Limit Theorem, as is indeed the=
case. A Poisson(1) distribution (see graph below) is quite skewed, so we w=
ould expect to need to add together some 20 or so before the sum would look=
approximately Normal.

*
*

The mean and variance of a Poisson(*l**t*) distribution are both equal to *l**t*. Thus, the Normal approxi=
mation to the Poisson is given by:

Poisson=
(*l**t*) =C2=BB<=
/span> Normal(*l**t*, (*l**t*)^{=
=C2=BD})

*l**t > 20*

A much more generally useful Normal approximation to th=
e Poisson distribution is given by the formula:

Poisson(*l*) =C2=BB [Normal(2*l*^{=C2=BD}, 1)/2]^{2<=
/sup>}

This formula works for values of *l as low as 1.*

*
*

The discrete property of the variable is lost with this=
approximation. The comments also apply here for retrieving the discretenes=
s and reducing error at the same time that are presented for the Normal approximation to the Binomial.