7. On the basis of a physical examination and symptoms, a physician assesses the probabilities that the patient has no tumour, a benign tumour, or

a malignant tumour as 0.70, 0.20, and 0.10, respectively. A thermographic

test is subsequently given to the patient. This test gives a negative result

with probability 0.90 if there is no tumour, with probability 0.80 if there is

a benign tumour, and with probability 0.20 if there is a malignant tumour.

a) What is the probability that a thermographic test will give a negative

result for this patient?

b) Obtain the posterior probability distribution for the patient when the

test result is negative?

c) Obtain the posterior probability distribution for the patient when the

test result is positive?

d) How does the information provided by the test in the two cases change

the physician's view as to whether the patient has a malignant tumour?

8. A small college has a five member economics department. There are two

microeconomists, two macroeconomists and one econometrician. The World

Economics Association is holding two conferences this year, one in Istanbul

and one in Paris. The college will pay the expenses of one person from the

department for each conference. The five faculty members have agreed to

draw two names out of a hat containing all five names to determine who

gets to go to the conferences. It is agreed that the person winning the trip

to the first conference will not be eligible for the draw for the second one.

a) What is the probability that the econometrician will get to go to a

conference?

b) What is the probability that macroeconomists will be the attendees at

both conferences?

6. A bright young economics student at Moscow University in 1950 criticized

the economic policies of the great leader Joseph Stalin. He was arrested and

sentenced to banishment for life to a work camp in the east. In those days

70 percent of those banished were sent to Siberia and 30 percent were sent

to Mongolia. It was widely known that a major difference between Siberia

and Mongolia was that fifty percent of the men in Siberia wore fur hats,

while only 10 percent of the people in Mongolia wore fur hats. The student

was loaded on a railroad box car without windows and shipped east. After58 PROBABILITY

many days the train stopped and he was let out at an unknown location.

As the train pulled away he found himself alone on the prairie with a single

man who would guide him to the work camp where he would spend the rest

of his life. The man was wearing a fur hat. What is the probability he

was in Siberia? In presenting your answer, calculate all joint and marginal

probabilities. Hint: Portray the sample space in rectangular fashion with

location represented along one dimension and whether or not a fur hat is

worn along the other.

Part A. For two independent events A and B, suppose that P(A) = 0.2 and P(B) = 0.4. To find P(A /7 B), what probability rule/law, should you use? O Axioms of probability O Definition of the independent events O The law of total probability O The multiplication rule for P(A / B) O Definition of the conditional probability O Bayes' theorem O The addition rule Find P(A /) B). To find P(A U B), what probability rule/law, should you use? O The multiplication rule for P(A / B) O Definition of the conditional probability O Bayes' theorem O The law of total probability O The addition rule O Definition of the independent events O Axioms of probability Find P(A U B).5. (35 points) Suppose that two random variables X and Y have a joint probability density function f(x, v) = / be (2:+3)/6 for z > 0 and y > 0, otherwise. (a) Are the two random variables X and Y independent? Explain. (b) Find the joint cumulative distribution function (cdf), Fx.y(u, v). (c) Find P(0 2 X = 1).The joint cumulative distribution function of the random variables Y1 and Y2 is given below. Find the joint probability density function and the marginal density functions of these two random variables. F(y1, '2)=1-e OSyISV2 O otherwiseProblem 2 (40 points). Continuous random variables X and Y are jointly distributed according to the PDF below. cy 73 x > 2,0 (0.5X - 1)? (f) Find the joint cumulative distribution function (CDF) for X and Y