Physics and Engineering of Radiation Detection

538 Chapter 9. Essential Statistics for Data Analysis

x

0 1020304050

f

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18 μ=5.5

μ=10

μ=15

μ=30

μ=20

Figure 9.3.3: Poisson probabil-

ity density for different values of

μ. The width of the distribu-

tion, a reflection of the uncer-

tainty in the measurement, in-

creases with increase inμ.

The log likelihood function ofL(μ)is

l≡ln(L)=

(n

∑

i=1

xi

)

ln(μ)−nμ−ln(x 1 !x 2 !...xn!). (9.3.33)

Following the maximum likelihood method (∂l/∂μ=0)weget

∂

∂μ

[(n

∑

i=1

xi

)

ln(μ)−nμ−ln(x 1 !x 2 !...xn!)

]

=0

1

μ∗

∑n

i=1

xi−n =0

μ∗ =

1

n

∑n

i=1

xi. (9.3.34)

This shows that the simple mean is the most probable value of a Poisson distributed
variable. To determine the error inμ, we fist take second derivative of the log
likelihood function and then substitute it in equation 9.3.22.

∂^2 l

∂μ^2

= −

1

μ^2

∑n

i=1

xi

μ =

[

−

∂^2 l

∂μ^2

]− 1 / 2

=

[

μ∗^2

∑n

i=1xi

] 1 / 2

=

1

n

[n

∑

i=1

xi

] 1 / 2

(9.3.35)

This is one of the most useful results of the Poisson distribution. It implies that if
we make one measurement, the statistical error we should expect in it would simply