Physics and Engineering of Radiation Detection

9.3. Probability 543

Suppose we havemindependent normally distributed random variablesuihaving
theoretical meansμiand variancesσi^2 .χ^2 is then defined as

χ^2 ≡x=

∑m

i=1

(ui−μi)^2

σi^2

. (9.3.53)

The parameternin the definition of theχ^2 probability distribution is then related to
the number of independent variables in this equation. For largentheχ^2 distribution
reduces to the Gaussian distribution with meanμ= nand varianceσ^2 =2n.
Fig.9.3.5 shows the shapes of the chi-square distribution for different degrees of
freedom. Asnincreases, the distribution assumes a shape that becomes more and
more like a Gaussian or normal distribution.

x

0 5 10 15 20 25

f

0

0.1

0.2

0.3

0.4

0.5

n=2

n=5

n=7 n=10

Figure 9.3.5: χ^2 probability den-

sity functions for different degrees

of freedomn. The shape of the

distribution approaches that of

a Gaussian distribution with in-

creasingn.

To understand the utility of this distribution function, let us have a closer look
at the definition 9.3.53. The numerator in this equation represents the deviations of
the normally distributed variableuifrom its theoretical means at each data point
while the denominator represents its expected standard deviations. In other words
the numerator and denominator are the actual and expected deviations respectively.
If the datauiis really Gaussian distributed, then ideally the actual deviation should
be equal to the expected one. However in real data there are always fluctuations
and consequently these deviations are not equal. Theχ^2 probability density function
9.3.52 actually tells us how the probability of this deviation is distributed. Hence
this distribution function can be used to judge the data against a hypothetical mean.
We will learn more about this when we discuss the goodness-of-fit tests later in the
chapter.

D.5 Student’stDistribution

Student’stdistribution is a widely used probability distribution. It forms the ba-
sis of Student’st-test, which we will discuss later in the chapter. To define this