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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 lambda, the probability of getting an event in an interval dx is dx / lambda. Let P(0, x) be the probability of getting no events in a length x. Then we have

Equation 19     (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:

Equation 20     (20)

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

Equation

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

Equation 21 the Poisson
distribution
    (21)

As a check, note

Equation

Likewise it can be shown that img84 = Nbar . Equation (21) is often expressed in terms of Nbar:

Equation 22 the Poisson
distribution
    (22)

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

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 alpha ident Nbar and its error:

Equation

Thus we have

Equation

and by Eq. (7),

Equation

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

Equation

In conclusion we note that alpha neq alphabar :

Equation

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