Physics and Engineering of Radiation Detection

9.3. Probability 545

Now that we have learned all the basics of maximum likelihood methodology, we
are ready to use it in practical situations. By practical situations we mean the real
cases corresponding to distributions that are not perfectly described by the standard
distribution functions we just studied. Let us suppose we have a variablex,whose
probability distribution function can be written as

p(x, k)=ke−kt,

wherekis a constant. Suppose we take 4 measurements of the parametert:t=
12 , 14 , 11 ,12. What we want to do is to use the maximum likelihood method to
compute the value of the constantk. To do this we first need to determine the
maximum likelihood function. According to equation 9.3.19, this is give by

L(k)=

∏^4

i=1

ke−kti

= e−k

(^4) (t 1 +t 2 +t 3 +t 4 )
= k^4 e−^49 k.
Now we take the natural logarithm of this function.
ln(L)=4ln(k)− 49 k
According to the maximum likelihood method, differentiating the above function
with respect tokand equation the result to zero gives the required maximum like-
lihood estimate ofk.
∂ln(L)
∂k

=0

⇒

4

k

−49 = 0

⇒k =

4

49

.

Let us now look at another example. This time we want to know how many
measurements we must make so that the parameterk=0.21 of the distribution

f(x, k)=kx ;x∈(0,1),

can be determined with an accuracy of 5%. That is, the relative error ink=0. 21
is
k
k

=0. 05. (9.3.56)

This can easily be done by using equation 9.3.51, which for our case becomes

N=

1

( k)^2

∫ 1

0

1

f

(

∂f

∂k

) 2

dx.