Physics and Engineering of Radiation Detection

9.3. Probability 537

Hence maximum of ln(L)atp∗is

r

p∗

+

N−r

1 −p∗

=0

⇒p∗ =

r

N

. (9.3.28)

Now in order to evaluate the error inp∗we differentiate again equation 9.3.27 with
respect to p to get
∂^2 ln(L)
∂p^2

=

r

p^2

−

N−r

(1−p)^2

(9.3.29)

According to equation 9.3.22, the error inp∗is then given by

p =

[

−∂^2 ln(L)

∂p^2

]− 1 / 2

=

[

r

p∗^2

+

N−r

(1−p∗)^2

]− 1 / 2

=

[

p∗(1−p∗)

N

] 1 / 2

, (9.3.30)

where we have usedr=p∗N.

D.2 PoissonDistribution

Poisson distribution represents the distribution of Poisson processes and is in fact
a limiting case of the Binomial distribution. By Poisson processes we mean the
processes that are discrete, independent and mutually exclusive.
The p.d.f. of a Poisson distribution is defined as

f(x;μ)=

μxe−μ

x!

, (9.3.31)

withx =0, 1 , .... represents thediscreterandom variable, such as ADC counts
obtained from a detection system andμ>0 is the mean. Fig.9.3.3 depicts this
distribution for different values ofμ. It is apparent that the width of the distribution
increases withμ, which indicates that the uncertainty in measurement increases with
an increase in the value ofx.
Let us now apply the maximum likelihood method to determine the best estimate
of mean of a set ofnmeasurements assuming that the underlying process is Pois-
son in nature. The best way to do this is to use the maximum likelihood method
we outlined earlier and applied in the previous section while discussing the Bino-
mial distribution. Since Poisson distribution is a discrete probability distribution
therefore its likelihood function for a set ofnmeasurements can be written as

L(μ)=

∏n

i=1

f(xi,μ)

=

∏n

i=1

[

μxie−μ

xi!

]

=

μ

Px

ie−nμ

x 1 !x 2 !...xn!

(9.3.32)