Physics and Engineering of Radiation Detection

540 Chapter 9. Essential Statistics for Data Analysis

x

0 5 10 15 20 25 30 35 40 45

f

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

x

-4 -2 0 2 4

f

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Figure 9.3.4: (a) Gaussian dis-

tribution forμ =25andσ =

5. (b) Standard normal distri-
   bution havingμ =0andσ =


6. With proper change of scale,
   any Gaussian distribution can be
   transformed into a standard nor-
   mal distribution.

Taking the natural logarithm of both sides of this equation gives

ln(L)=

∑N

i=1

[

−

(xi−μ)^2

2 σ^2 i

−ln(σi)−

ln(2π)

2

]

. (9.3.40)

The maximum likelihood solution is then obtained by differentiating this equation
with respect toμand equating the result to zero. Hence we get

∂ln(L)

∂μ∗

=

∑N

i=1

xi−μ∗

σ^2 i

= 0 (9.3.41)

⇒

∑N

i=1

μ∗

σi^2

=

∑N

i=1

xi

σi^2

⇒μ∗ =

∑N

∑i=1wixi

N

i=1wi

, (9.3.42)

wherewi=1/σi^2. Hence the most probable value is simply the weighted mean with
respective inverse variances orerrorsas weights. If we assume that each measure-
ment has the same amount of uncertainty or error then usingσi=σthe maximum