### 13. POISSON DISTRIBUTION

A common type of problem which falls into this category is the determination of a cross section or a mean free path. For a mean free path , the probability of getting an event in an interval dx is dx / . Let P(0, x) be the probability of getting no events in a length x. Then we have

 (19)

Let P(N, x) be the probability of finding N events in a length x. An element of this probability is the joint probability of N events at dx1, ..., dxN times the probability of no events in the remaining length:

 (20)

The entire probability is obtained by integrating over the N-dimensional space. Note that the integral

does the job except that the particular probability element in Eq. (20) is swept through N! times. Dividing by N! gives

 the Poisson distribution (21)

As a check, note

Likewise it can be shown that = . Equation (21) is often expressed in terms of :

 the Poisson distribution (22)

This form is useful in analyzing counting experiments. Then the "true" counting rate is .

We now consider the case in which, in a certain experiment, N events were observed. The problem is to determine the maximum-likelihood solution for and its error:

Thus we have

and by Eq. (7),

In a cross-section determination, we have = x , where is the number of target nuclei per cm3 and x is the total path length. Then

In conclusion we note that :