### DEPARTMENT OF MATHEMATICS & STATISTICS STAT 3083

 December 1996 Final Examination Time: 3 Hours
1. A test has been developed to detect a particular type of arthritis in individuals over 60 years old. From a national survey it is known that approximately 15% of the individuals in this group suffer from this form of arthritis. The proposed test was given to confirmed arthritic disease, and correct test results were obtained in 80% of the cases. When the test was administered to individuals of the same age group who were known to be free of the disease, 4% were reported to have the disease. What is the probability that an individual has this disease given that the test indicated it is present?

2. Let T be a discrete random variable with its probability density function given by the following:

 T 1 2 3 4 5 f(t) 0.15 0.18 0.27 0.22 0.18

 (a) Find E(T), Var(T). (b) Find P[T > 3 | T > 2] (c) Find E(T | T > 2) (d) Find the values of the cumulative distribution function: F(0), F(1), F(2), F(3), F(4), F(5), F(6)

3. An owner of five overnight cabins is considering buying television sets to rent to cabin occupants. He estimates that about half his customers would be willing to rent sets. Finally, he buys three sets. Assuming 100% occupancy at all times:
 (a) what fraction of evenings will there be more requests than TV sets; (b) if the owner charges \$2.00 rental fee per set per evening, then what will be his expected rental revenue?

4. Let Y possess a density function

 (a) Find c. (b) Find the cumulative distribution function, F(y). (c) Find P[Y < 1 | Y < 1.5]. (d) Find E(Y | Y < 1.5). (e) Find the hazard rate function, , 0 < t < 2.

5. Life of certain electronic components, X, obey exponential model with mean life of 11 hours.
 (a) Find the 'half-life' of such components, i.e. find 'b' such that P[ X > b] =1/2. (b) What is the expected life of a 6 hours old component, E( X | X > 6) ? (c) What is the probability that a 6 hours old component will survive another five hours,   P[ X > 11 | X > 6] ?

6. Suppose that the scores on a certain test are normal with mean 120 and standard deviation 30.
 (a) What proportion of scores are over 100? (b) Find the 57th percentile, i.e. the best score in the low 57% group.

7. Find the mean and variance of a random variable having its hazard function, , given by the following

= 6 t,                 t > 0.

8. Let X and Y be two jointly distributed random variables with the following joint distribution.

Y
X
1 2 3 4
2 0.12 0.08 0.15 0.05
4 0.07 0.06 0.12 0.05
6 0.06 0.04 0.05 0.00
8 0.05 0.02 0.08 0.00
(a) Are X, Y, independent? Explain.
(b) Find P[X > 3, Y > 3]
(c) Find Cov(X, Y)
(d) Find E(X2Y | X > 4 Y > 3]
(e) Find P[X+Y < 5].

9. Let X1, X2, ... be i.i.d. random variables with mean = 36 and standard deviation = 5, and let Y1, Y2, ... be i.i.d. random variables with mean = 30 and standard deviation = 5. Let

W = X1+X2+...+X40+ Y1+Y2+...+Y40 .

Find P[W < 2600].

10. Let X1, X2, ... be i.i.d. random variables having the density functiion of a chi-square variable with 3 degrees of freedom. Let

W=X1+X2+...+X30 .
 (a) Find P[W < 90]. (b) Find P[W > 100].