Both the binomial and Poisson distributions apply to discrete variables, whereas most of the random variables involved in experiments are continuous. In addition, the use of discrete distributions necessitates the use of long or infinite series for the calculation of such parameters as the mean and the standard deviation (see Eqs. 2.47, 2.48, 2.52, 2.53). It would be desirable, therefore, to have a pdf that applies to continuous variables. Such a distribution is the normal or Gaussian distribution.
The normal distribution G ( x ) is given by
where G(x) dx = probability that the value of x lies between x and x + &
m = average of the distribution a ' = variance of the distribution
Notice that this distribution, shown in Fig. 2.4, has a maximum at x = m, is symmetric around m, is defined uniquely by the two parameters u and m, and extends from x = - m to x = + m . Equation 2.55 represents the shaded area under the curve of Fig. 2.4. In general, the probability of finding the value of x
40 MEASUREMENT AND DETECTION OF RADIATION
Figure 2 3 Three Poisson distributions: (a) rn = 5, ( b ) rn = 10, ( c ) rn = 20.
0 m - o m m + o x x + d x X
Figure 2.4 A normal (Gaussian) distribution.
between any two limits x , and x2 is given by
The
The
Gaussian given by Eq. 2.55 satisfies
average of the distribution is
The variance is
The standard deviation is
Three very important items associated with the Gaussian distribution are the following.
1. The cumulative normal distribution function, defined by
x 1 (x' - m )
E ( x ) = G ( x t ) dx' =
- m EX^[- 2 u 2 ] &' (2.61)
42 MEASUREMENT AND DETECTION OF RADIATION
The function E(x) is very useful and is generally known as the error function (see also Sec. 2.10.1). Graphically, the function E(x) (Eq. 2.61) is equal to the shaded area of Fig. 2.5. The function is sketched in Fig. 2.6.
2. The area under the curve of Fig. 2.4 from x = m - u to x = m + u , given by
Equation 2.62 indicates that 68.3 percent of the total area under the Gaussian is included between m - u and m + u . Another way of expressing this statement is to say that if a series of events follows the normal distribution, then it should be expected that 68.3 percent of the events will be located between m - u and m + a. As discussed later in Sec. 2.13, Eq. 2.62 is the basis for the definition of the "standard" error.
3. The full width at half maximum (FWHM). The F W H M , usually denoted by the symbol r , is the width of the Gaussian distribution at the position of half of its maximum. The width r is slightly wider than 2 u (Fig. 2.4). The correct
Figure 2.5 The cumulative normal distribution is equal to the shaded area under the
0 x m x' Gaussian curve.
relationship between the two is obtained from Eq. 2.55 by writing
Solving this equation for r gives
The width r is an extremely important parameter in measurements of the energy distribution of particles.
2.10.1 The Standard Normal Distribution
The evaluation of integrals involving the Gaussian distribution, such as those of Eqs. 2.56, 2.61, and 2.62, requires tedious numerical integration. The result of such integrations is a function of m and u. Therefore, the calculation should be repeated every time m or u changes. To avoid this repetition, the normal distribution is rewritten in such as way that
m = O and u = 1
The resulting function is called the standard normal distribution. Integrals involving the Gaussian distribution, such as that of Eq. 2.61, have been tabu- lated based on the standard normal distribution for a wide range of x values.
With the help of a simple transformation, it is very easy to obtain the integrals for any value of m and u.
The standard normal distribution is obtained by defining the new variable.
x - m t = -
u Substituting into Eq. 2.55, one obtains
It is very easy to show that the Gaussian given by Eq. 2.65 has mean
and variance
The cumulative standard normal distribution function, Eq. 2.61, is now written as
44 MEASUREMENT AND DETECITON OF RADIATION
or, in terms of the error function that is tabulated
where
Example 2.12 The uranium fuel of light-water reactors is enclosed in metallic tubes with an average outside diameter (OD) equal to 20 mm. It is assumed that the OD is normally distributed around this average with a standard deviation a = 0.5 mm. For safety reasons, no tube should be used with
OD > 21.5 mm or OD < 18.5 mm. If 10,000 tubes are manufactured, how many
of them are expected to be discarded because they do not satisfy the require- ments given above?
Answer The probability that the OD of a tube is going to be less than 18.5 mm or greater than 21.5 mm is
Graphically, the sum of these two probabilities is equal to the two shaded areas shown in Fig. 2.7.
In terms of the standard normal distribution and also because the two integrals are equal, one obtains
where
x - 20
t = ---
0.5
This last integral is tabulated in many books, handbooks, and mathematical tables (see bibliography of this chapter). From such tables, one obtains
which gives
G(x < 18.5) + G(x > 21.5) = 0.0027
Figure 2.7 The shaded areas rep- resent the fraction of defective rods, Ex. 2.12.
Therefore, it should be expected that under the manufacturing conditions of this example, 27 tubes out of 10,000 would be rejected.
2.10.2 Importance of the Gaussian Distribution for Radiation Measurements
The normal distribution is the most important distribution for applications in measurements. It is extremely useful because for almost any type of measure- ment that has been taken many times, the frequency with which individual results occur forms, to a very good approximation, a Gaussian distribution centered around the average value of the results. The greater the number of trials, the better their representation by a Gaussian. Furthermore, statistical theory shows that even if the original population of the results under study does not follow a normal distribution, their average does. That is, if a series of measurements of the variable xili= I , . . . , N is repeated M times, the average values Z N I N = . . , follow a normal distribution even though the xi's may not.
This result is known as the central limit theorem and holds for any random sample of variables with finite standard deviation.
In reality, no distribution of experimental data can be exactly Gaussian, since the Gaussian extends from - w to +m. But for all practical purposes, the approximation is good and it is widely used because it leads to excellent results.
It is worth reminding the reader that both the binomial (Fig. 2.2) and the Poisson (Fig. 2.3) distributions resemble a Gaussian under certain conditions.
This observation is particularly important in radiation measurements.
The results of radiation measurements are, in most cases, expressed as the number of counts recorded in a scaler. These counts indicate that particles have interacted with a detector and produced a pulse that has been recorded. The particles, in turn, have been produced either by the decay of a radioisotope or as a result of a nuclear reaction. In either case, the emission of the particle is statistical in nature and follows the Poisson distribution. However, as indicated in Sec. 2.9, if the average of the number of counts involved is more than about 20, the Poisson approaches the Gaussian distribution. For this reason, the
46 MEASUREMENT AND DETECTION OF RADIATION
individual results of such radiation measurements are treated as members of a normal distribution.
Consider now a Poisson and a Gaussian distribution having the same average, m = 25. Obviously, there is an infinite number of Gaussians with that average but with different standard deviations. The question one may ask is:
"What is the standard deviation of the Gaussian that may represent the Poisson distribution with the same average?" The answer is that the Gaussian with a
= d k= 5 is almost identical with the Poisson. Table 2.1 presents values of the two distributions, and Fig. 2.8 shows them plotted.
The following very important conclusion is drawn from this result:
The outcomes of a series of radiation measurements are members of a Poisson distribution.
They may be treated as members of a Gaussian distribution if the avera e result is more than m = 20. The standard deviation of that Gaussian distribution is u = P m .
Use of this conclusion is made in Sec. 2.17, which discusses statistics of radiation counting.