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 :