Physics and Engineering of Radiation Detection

9.3. Probability 539

be the square root of the measured quantity. For example, if we count the number of
γ-ray photons coming from a radioactive source using a GM-tube and get a number
N, the statistical error we should expect will simply be

√

N. Fortunately, most of
the processes we encounter in the field of radiation detection and measurement, such
as activity of a radioisotope, photoelectric effect, and electron multiplication in a
PMT tube, can all very well be described by Poisson statistics.

D.3 NormalorGaussianDistribution

The normal distribution was originally developed as an approximation to the bi-
nomial distribution. Its usefulness was soon recognized by scientists and soon it
became one of the most commonly used probability distributions in not only these
fields but also in other sciences. The utility of normal distribution can be appre-
ciated by noting an amazing property of most of the physical processes that their
random variables can be safely approximated to be distributed normally. Therefore
a common practice is to assume that a random variable having unknown distribu-
tion can be defined by a normal distribution. This property of random variables
is actually the result of the so calledcentral limit theorem, which states that the
mean of any set of variables with any distribution tend to the normal distribution
provided their mean and variance are finite. Although the term normal distribution
is very commonly used, still some scientists prefer to call it Gaussian distribution.
Gaussian distribution has a bell-shaped curve (see Fig.9.3.4) and is defined for a
variablexin the domainx∈(−∞,∞)by

f(μ;x)=

1

σ

√

2 π

e−(x−μ)

(^2) / 2 σ 2
, (9.3.36)
whereμandσare the mean and standard deviation of the distribution respectively.
Bothμandσare finite for a normally distributed variable.
If we substituteμ=0andσ^2 = 1 in the above equation, we obtain the so called
standard normal distributionwith a probability density function given by
P(x)=

1

√

2 π

e−x

(^2) / 2

. (9.3.37)

This is simply a special kind of Gaussian distribution having a symmetric bell shaped
curve centered atx= 0 (see Fig.9.3.4). In fact by changing the variables, any
normal distribution can be easily converted into a standard normal distribution (see
Example below).
Let us now apply our maximum likelihood method to compute the most probable
value and its accuracy assuming the variable to be Gaussian distributed. Suppose
we makeNmeasurements of a variable and represent the result byxi.Eachofthese
measurements will have its own errorσi. Then according to equation 9.3.19, the
likelihood function is given by

L(μ)=

∏N

i=1

1

σi

√

2 π

e−(xi−μ)

(^2) / 2 σ (^2) i
(9.3.38)
In order to apply the condition 9.3.23, we rewrite the above equation in the form
L(μ)=

∏N

i=1

(e−(xi−μ)

(^2) / 2 σi 2
)(σ−i^1 )(2π)−^1 /^2 (9.3.39)