Physics and Engineering of Radiation Detection

9.11. Counting Statistics 571

Problems

1.Show that for large number of data points, the maximum likelihood function

approaches the Gaussian distribution function.

2.Using maximum likelihood method show that the number of measurements

needed to get a particular value of the parameterξwith a certain accuracy can

be obtained by

N=

1

( ξ)^2

∫

1

L

(

∂L

∂ξ

) 2

dx,

whereLis the likelihood function. The accuracy of the measurement is char-

acterized by ξ. (Hint: Start by writing equation 9.3.49 forNnumber of

measurements.)

3.In a radioactive decay experiment for measuring activity of a long lived sample,

the following counts are observed:

2012, 2154, 1993, 2009, 2028, 2129

What would be your Maximum Likelihood function? (Hint: radioactive

decay can be characterized as a Poisson process.)


Estimate the value of the most probable activity (counts) using the max-

imum likelihood method.

4.For the function of the previous exercise, estimate how many measurements

must be performed to get a value ofμ=0.5 with 1% accuracy.

5.Given the following ADC counts observed in a radioactive decay counting ex-

periment, perform the Student’sttest to determine if the means of the two

datasets differ significantly from each other.

Measurement-1: 200, 220, 189, 204, 199, 201, 217

Measurement-2: 190, 230, 179, 188, 218

6.In a research paper the following experimental results of two separate measure-

ments of a quantity are given.

112. 56 ± 0. 78

104. 23 ± 0. 34

If the number of data points for each measurement was 12, determine at 95%

confidence level if the two means are significantly different from each other.

7.In two different experiments, the half life of a radioactive sample is found to

be 15. 5 ± 2 .3daysand16. 2 ± 1 .5 days. Determine the best estimate of the half

life by combining the two results.

8.In a radiation hardness study of a silicon detector, the system is exposed to

a constant flux of radiation over a long period of time. The damage caused

by the integrated flux is measured by noting the leakage current at regular

intervals of time. The following data are obtained (units are arbitrary)