Subatomic Physics

4.2. Statistical Aspects 57
Or, to make it more specific, consider a process where the average number of output
electrons is small, say n =3.5. What is the probability of finding the value n =2?
This problem has occupied mathematicians for a long time, and the answer is well
known (1) : The probability P (n) of observing n events is given by the Poisson
distribution,
P (n) = (n)n
n! e−n , (4.3)
where n is the average defined by Eq. (4.2). As behooves a probability, the sum
over all possible values n is 1, �∞ n=0 P (n) = 1. With Eq. (4.3), the previous
questions can now be answered, and we first turn to the most specific one. With
n =3.5,n = 2, Eq. (4.3) gives P (2) = 0.185. It is straightforward to compute the
probabilities for all interesting values of n. The corresponding histogram is shown
in Fig. 4.5. It shows that the distribution is very wide. There is a nonnegligible
probability of measuring values as small as zero or as large as 9. If we perform only
one measurement and find, for instance, a value of n = 7, we have no idea what the
average value would be.
A glance at Fig. 4.5 shows that it is not enough to measure and record the
average, n. A measure of the width of the distribution is also needed. It is customary
to characterize the width of a distribution by the variance σ2 :
σ 2 ∞�
= (n − n) 2 P (n), (4.4)
or by the square root of the variance, called the standard deviation.
For the Poisson distribution, Eq. (4.3), variance
and standard distribution are easy to
compute, and they are given by
P(n)
n=3.5
n=0
σ 2 = n, σ = √ n. (4.5)
For small values of n, the distribution is
not symmetric about n, asisevidentfrom
Fig. 4.5.
So far we have discussed the Poisson distribution
for small values of n. Experimentally,
such a situation arises, for instance, at the
first dynode of a photomultiplier, where each
incident electron produces two to five secondary
electrons. Data are then given in the
form of histograms, as in Fig. 4.5.
0.2
0.1
2�
0 1 2 3 4 5 6 7 8 9 n
Figure 4.5: Histogram of the Poisson
distribution for n =3.5. The distribution
is not symmetric about n.
1 A derivation can, for instance, be found in H. D. Young, Statistical Treatment of Experimental
Data, McGraw-Hill, New York, 1962, Eq.(8.5); R.A. Fisher, Statistical Methods, Experimental
Design, and Scientific Inference, Oxford Univ. Press, Oxford, 1990.