Physics and Engineering of Radiation Detection

544 Chapter 9. Essential Statistics for Data Analysis

distribution, let us first write

z =

∑n

i=1

x^2 i

and t =

x

√

z/n

,

fornindependent Gaussian variables having 0 mean and 1 variance. The variable
zin this expression follows theχ^2 -distribution we defined above and the variablet
follows Student’stdistribution with n degrees of freedom defined by

f(t;n)=

1

√

nπ

Γ[(n+1)/2]

Γ(n/2)

[

1+

t^2

n

]−(n+1)/ 2

, (9.3.54)

where Γ is the familiar gamma function, the variabletcan take any value (−∞<
t<∞), andncan be a non-integer.
The Student’stdistribution looks very similar to Gaussian distribution. For
smalln, however it has wider tails, which approach that of a Gaussian distribution
with increasingn.

x

-4 -2 0 2 4

f

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

n=2

n=30

Figure 9.3.6: Student’stdistri-

bution for two values of degrees

of freedom n.Asn increases

the tails of the distribution ap-

proaches that of a Gaussian dis-

tribution.

D.6 GammaDistribution......................

For a Poisson process the distance inxfrom any starting point to thekthevent
follows Gamma distribution given by

f(x;λ, k)=

xk−^1 λke−λx

Γ(k)

, (9.3.55)

with 0<t<∞andkcan be a noninteger.
Forλ=1/2andk=n/2 it reduces to theχ^2 -distribution we defined above.

Using Maximum Likelihood Method