EXERCISES 3 5

there. Or you may get tails and I get heads, or we may both get tails. In two of the three possible cases, you win." "Right," says your friend, "I have two chances to your one. So Ill put up $2 against $1." Is this a fair bet?

3. A major investment company wanted to know what proportions of investors bought stocks, bonds, both stocks and bonds, and neither stocks nor bonds. The company entrusted this task to its statistician, who, in turn, asked her assistant to conduct a telephone survey of 200 potential investors chosen at random. The assistant, however, was a moonlighter with some preconceptions of his own, and he cooked up the following results without actually doing a survey. For a sample of 200, he reported the following results:

Invested in stocks: 100

Invested in bonds: 75

Invested in both stocks and bonds: 45

Invested in neither: 90

The statistician saw these results and fired his moonlighting assistant. Why?

4. (The birthday problem) (In this famous problem it is easier to calculate the probability of a complementary event than the probability of the given event.) Suppose your friend, a teacher, says that she will pick 30 students at random from her school and offer you an even bet that there will be at least two students who have the same birthdays. Should you accept this bet?

5. (Getting the answer without being sure you have asked the question) This is the randomized response technique due to Stanley L. Warners paper in Journal of the American Statistical Association, Vol. 60 (1965), pp. 63-69. You want to know the proportion of college students who have used drugs. A direct question will not give frank answers. Instead, you give the students a box containing 4 blue, 3 red, and 4 white balls. Each student is asked to draw a ball (you do not see the ball) and abide by the following instructions, based on the color of the ball drawn:

Blue: Answer the question: "Have you used drugs?" Red: Answer "yes." White: Answer "no."

If 40% of the students answered "yes," what is the percentage of students who have used drugs?

6. A petroleum company intends to drill for oil in four different wells. Use the subscripts 1, 2, 3,4 to denote the different wells. Each well is given an equally likely chance of being successful. Consider the following events:

Event A: There are exactly two successful wells. Event B: There are at least three successful wells. Event C: Well number 3 is successful. Event D: There are fewer than 3 successful wells.

(a) Are X and Y independent? Explain.

(b) Find the marginal distributions of X and F.

(c) Find the conditional distribution of F given X = 1 and hence E{Y\ X = l)andvar(FZ = 1).

(d) Repeat part (c) ior X = 2 and = 3 and hence verify the result that V(F) = £V(F I A) + VE{Y \ X) that is the variance of a random variable is equal to the expectation of its conditional variance plus the variance of its conditional expectation.

10. The density function of a continuous random variable X is given by

f( \ = ~ for 0 < : < 2

0 otherwise

Compute the following probabilities:

(a) P{A) (b) P{B) (c) PiC) (d) P{D)

(e) P{AB) (f) + B) (g) ( ) (h) ( + )

(i) F(fiD) (j) nC) (k) ( ) (1) ( )

(m) P{C\D) (n) F(£>C)

7. On the TV show "Lets Make a Deal," one of the three boxes (A, B, C) on the stage contains keys to a Lincoln Continental. The other two boxes are empty. A contestant chooses box B. Boxes A and remain on the table. Monty Hall, the energetic host of the show, suggests that the contestant surrender her box for $500. The contestant refuses. Monty Hall then opens one of the remaining boxes, box A, which turns out to be empty. Monty now offers $1000 to the contestant to surrender her box. She again refuses but asks whether she can trade her box for box on the table. Monty exclaims, "Thats weird." But is it really weird, or does the contestant know how to calculate probabilities? Hint: Suppose that there are n boxes. The probability that the key is in any of the (n - 1) boxes on the stage = ( - \)ln. Monty opens p boxes, which turn out to be empty. The probability that the key is in any of the (n - p - 1) boxes is ( - 1)/ n{n - p -1). Without switching, the probability that the contestant wins remains at l/n. Hence it is better to switch. In this case = 3, p = 1. The probability of winning without a switch is i The probability of winning by switching is §.

8. You are given 10 white and 10 black balls and two boxes. You are told that your instructor will draw a ball from one of the two boxes. If it is white, you pass the exam. If it is black, you fail. How should you arrange the balls in the boxes to maximize your chance of passing?

9. Suppose that the joint probability distribution of X and Y is given by the following table.

fix, y) = [

fl. - - >(2 - ) for < X < 1, < < 2 • - [ otherwise

14. Let dependent random variables X, Y, and Z be defined by the joint distribution:

P(Z = 1, F= 2,Z = 3) = 0.25 P(Z = 2, F = 1, Z = 3) = 0.35 P(Z = 2, F = 3, Z = 1) = 0.40

In this case, { < F) = 0.65, P{X < Z) = 0.6, and P(F< Z) = 0.6, which shows that Z is the largest. However, direct observation shows that

P{X = minimum ofX, Y, Z) = 0.25 PiY = minimum ofX, Y, Z) = 0.35 P(Z = minimum of F, Z) = 0.40

which shows that Z is most likely to be the smallest. What can you conclude? (See C. R. Blyth, Journal of the American Statistical Association, Vol. 67, 1972, pp. 364-366 and 366-381.)

15. The number of hot dogs sold at a football stand has the following probability distribution:

(a) Find A:.

(b) Find E{X) and V{X).

11. Answer Exercise 10 when

kx for 0 s X < 1

fix) = ] k{2) - x) for 1 < X < 2 0 otherwise

12. Suppose that X and Fare continuous random variables with the joint probability density function

for0x< l,0<y 2 otherwise

(a) Find k, E{X), E{Y), V{X), ViY), and cov(Z, Y). Are X and independent?

(b) Find the marginal densities of X and Y.

(c) Find the conditional density of X given = 5 and hence E(X Y = i)andViY\X = I).

13. Answer Exercise 12 if/(x, y) is defined as follows: