Physics and Engineering of Radiation Detection

542 Chapter 9. Essential Statistics for Data Analysis

9.3.41 again with respect toμ∗and substituting the result in the above expression.

∂^2 L

∂μ^2

= −

∑N

i=1

1

σ^2 i

⇒

1

σt^2

=

∑N

i=1

1

σi^2

(9.3.48)

This expression represents thelaw of combination of errors, which states that for
repeated measurements of a normally distributed variable having errorsσi,thein-
verse of the total error in the calculation of mean is equal to the sum of inverse of
individual measurement errors.
Unfortunately, in a number of practical problems, an analytic determination of
μis not possible. In such cases one tries to find the value of the likelihood function
at each point by iteratingμ(or more accurately, by trying different values ofμ).
The points thus obtained are then plotted and the likelihood function is obtained
by performing the best fit through the points. In most cases with large number of
data points the likelihood function is Gaussian like. If it isn’t, one must perform a
weighted average to determine the error function, that is

〈

∂^2 ln(L)

∂μ^2

〉

=

∫∂ (^2) ln(L)
∫∂μ^2 Ldμ
Ldμ

. (9.3.49)

This is an important relation since it can be used to show that (see problems at the
end of the chapter) the maximum likelihood error inμcan be evaluated from

μ=

[

1

N

∫

1

L

(

∂L

∂μ

) 2

dx

] 1 / 2

, (9.3.50)

whereN is the number of measurements. An interesting aspect of this result is
that it allows one to determine the number of measurements necessary to obtain a
particular value of the parameterμwith a certain accuracy, that is

N=

1

( μ)^2

∫

1

L

(

∂L

∂μ

) 2

dx. (9.3.51)

D.4 Chi-Square (χ^2 )Distribution

χ^2 -distribution is one of the most extensively used probability distributions to per-
form goodness-of-fit tests, which we will discuss later in the chapter. It is defined as

f(x;n)=

xn/^2 −^1 e−x/^2

2 n/^2 Γ(n/2)

, (9.3.52)

where in order to avoid confusion due to the exponent 2 ofχ^2 we have represented it
byx. Γ() is thegamma functionandx≥0. The tables as well as analytical forms of
gamma functions can be found in standard texts of statistics and mathematics. The
parameternin the above definition is called thedegrees of freedomof the system.
The meaning of this term can be understood by looking at the definition ofxorχ^2.