Problems

01

Identifying Uranium

You are testing gram samples of pure Uranium to see if they are enriched. You have a Geiger counter that counts a number of gamma rays that come from nearby fission events in 1 second intervals after you press the count button.

If the sample is enriched, you expect a Poisson distribution $N$ of gamma rays in the counter with an average of 20. If the sample is not enriched (the null hypothesis), the average count will be 10.

(a) Design an ML test to decide whether it is ordinary $(H_0)$ or enriched ($H_1$). What is $A_0$? What are the probabilities of Type I, Type II, and Total error?

(b) After running the test many times, you have noticed that 70% of the samples are ordinary, while 30% are enriched. Now design an MAP test. What is $A_0$? What are the probabilities of Type I, Type II, and Total error?

(c) Missing a bit of enriched Uranium is obviously a major problem. The damage to your reputation and pocketbook of missing enriched Uranium is $100\times$ the damage caused by incorrectly labeling ordinary Uranium as enriched. Now design an MC test. What is $A_0$? What are the probabilities of Type I, Type II, and Total error?

(d) What is the expected cost of each application of the MC test, assuming the cost of a false alarm is $10,000? What is this number for the MAP test?

02

Light bulbs

Light bulbs from box $A$ (the null hypothesis) typically last $1000\mathrm{hrs}$, and bulbs from box $B$ last $500\mathrm{hrs}$. You have some bulbs but don’t know which box they came from. Bulb lifetimes are exponential.

It costs $50 in processing if you mistakenly assign a $B$ bulb to box $A$, and $20 if you assign an $A$ bulb to box $B$.

After working at this for a while, you observed that 60% of the bulbs you see come from box $A$, and the rest from box $B$.

Design a binary hypothesis test using MC design to make a decision rule to assign bulbs to boxes.

(a) What is $A_0$?

(b) What are the probabilities of Type I, Type II, and Total error?

(c) What is the expected cost for each application of the test?

03

Security screening

A metal detector for an event produces a reading, $X$, that varies between 0 and 10 according to the PDFs given below. (Note $X$ is a continuous random variable.)

Based on the reading, a security guard will stop and search a person or let them pass. Suppose it is known that 10% of people passing through security are carrying metal objects.

$H_0=$ a person is not carrying metal objects $H_1=$ a person is carrying metal objects

$$\begin{array}{cc}{f}_{X|H_0}(x)& =\frac{10-x}{50},0<x<10\\ & \\ f_{X|H_1}(x)& =\frac{x}{50},0<x<10\end{array}$$

Suppose it is 20 times worse to neglect searching someone who is carrying metal than to search someone who is not carrying metal. Design a minimum cost test that uses the value of the reading, X to decide whether the security guard will stop that person. Clearly state the decision rule.

04

Medical testing

A doctor is planning to use a new, inexpensive medical test to detect a particular disease. The test score, $X$, tends to be higher for patients with the disease. The PMFs for the test score for patients with and without the disease are shown below. From a previously used, more expensive test, it is known that 20% of the population has this disease.

Patients without the disease:

$x$	1	2	3	4	5
$P_X(x)$	0.5	0.3	0.15	0.05	0

Patients with the disease:

$x$	1	2	3	4	5
$P_X(x)$	0.05	0.1	0.3	0.35	0.2

Design a binary hypothesis test that will minimize the doctor’s probability of error. Let $H_0$: the patient does not have the disease and $H_1$: the patient does have the disease. Determine for which test scores the doctor should diagnose the patient as having the disease. Clearly denote which scores result in which decisions.

05

CAT scan for tumors

When a brain is scanned in a CAT scan, analysis of the results yields a rating of 1, 2, 3, or 4. This represents (imperfect) evidence of whether there is a tumor.

	$X$	1	2	3	4
No tumor:	$P_X(k)$	0.4	0.3	0.2	0.1
	$X$	1	2	3	4
With tumor:	$P_X(k)$	0.0	0.1	0.3	0.6

Suppose that, of people who get CAT scans, 20% do have a tumor.

Furthermore, assume that declaring there is no tumor when there is one is ten times worse than declaring there is a tumor when there isn’t one.

Design an MC test to determine which ratings should be classified as tumors.