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
*dx*_{1}, ..., *dx*_{N} 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 cm^{3} and *x* is the total
path length. Then

In conclusion we note that :