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# Statistics 5101, Fall 2002 (Geyer) Homework Assignments

Go to assignment:     1     2     3     4     5     6     7     8     9     10     11     12

No. Due Date Sec. Exercises Comments
1 Fri Sep 13 1.4 3, 6
1.5 2, 4, 6
1.6 2, 6
1.7 6, 10
1.8 4, 10
1.9 4
2 Fri Sep 20 2.1 2, 4, 6
2.2 3, 4, 6
2.3 2, 8, 9
3 Fri Sep 27 3.1 2, 4, 8
3.2 2, 4, 8
3.3 2, 4, 8
A 1, 2 "additional problems" see below.
4 Fri Oct 4 3.4 2, 4, 6
3.5 2, 6, 8
A 3 "additional problems" see below.
5 Wed Oct 16 3.6 2, 4, 10
3.7 4, 6 note: prob 2, originally assigned, deleted.
3.8 2, 4, 5, 8 in prob 5 the left hand side should be g(y), that is, little g of little y.
6 Wed Oct 23 3.9 4
3.10 21, 22
4.1 2, 6, 8
4.2 2, 4, 6, 8
7 Wed Oct 30 4.3 4, 5, 6, 9
4.4 8, 10
3.1 11 Not a mistake, going back to 3.1 for a problem about existence of infinite sums.
3.2 9 Not a mistake, going back to 3.2 for a problem about existence of integrals.
4.3 8 Not a mistake, going back to 4.3 for a problem about existence of integrals.
A 4, 5 "additional problems" see below.
8 Wed Nov 6 4.5 2, 3, 6, 12
4.6 5, 6, 10, 14 note Q5 added late.
9 Wed Nov 20 4.7 11, 12
A 6, 7 "additional problems" see below.
4.8 9
5.2 4, 8, 12
5.3 6
5.4 6, 8, 11
A 8, 9 "additional problems" see below.
10 Wed Nov 27 5.5 2, 6, 7
5.6 3, 4, 10, 14
11 Fri Dec 6 5.7 2, 6, 13, 14 probabilities calculated in 2 and 6 are only approximate.
5.8 2, 6
5.9 1, 4, 6, 10, 12
12 Fri Dec 13 5.10 2, 10
5.11 2, 4, 6
5.12 2, 4, 6

1. Suppose the probability density function f of a random variable X is defined by

f(x) = 1 / x2,     x > 1
f(x) = 0,     x < 1
(the value of f at x = 1 does not matter).

Find the cumulative distribution function of X.

2. Suppose the cumulative distribution function F of a random variable X is defined by

F(x) = 0,     x < 0
F(x) = x4,     0 ≤ x ≤ 1
F(x) = 1,     x > 1

Find the probability density function of X.

3. Suppose the joint probability density function f of random variables X and Y is defined by

f(x, y) = 2 y exp(- x y - 2 y),     x > 0 and y > 0
f(x, y) = 0,     otherwise

1. Find the marginal probability density function of X.
2. Find the marginal probability density function of Y.
3. Are X and Y independent random variables?

4. Suppose X has probability density

f(x) = 1 / (2 &radic x),     0 < x < 1
(note the domain).
1. For what positive integers k does Xk have expectation?
2. Calculate E(Xk) for the positive integers k such that the expectation exists.

5. Suppose X has probability density

f(x) = c / (ν + x2)(ν + 1) / 2,     - ∞ < x < + ∞
where c is a positive constant and ν is another positive constant.
1. Show that f is integrable (so some positive c exists that makes f a probability density).
2. For what positive integers k does E(Xk) exist? (This depends on the value of ν.)

6. Suppose X, Y, and Z are random variables such that

E(X | Y, Z) = Y
Var(X | Y, Z) = Z

Find the (unconditional) mean and variance of X in terms of the means, variances, and covariance of Y and Z.

(Hint: You must use the iterated conditional mean and variance formulas: Theorem 4.7.1 and Problem 4.7.11 in DeGroot and Schervish.)

7. Do Problem 4.7.15 in DeGroot and Schervish, but instead answer the following questions

1. If a student's score on the mathematics test is x, what predicted value of his or her score on the music test has the smallest M. S. E.? [Same as part (a) of the question in DeGroot and Schervish, except the number 0.8 is replaced by the variable x. Of course, plugging in x = 0.8 gives the answer in the back of the book.]
2. If a student's score on the music test is y, what predicted value of his or her score on the music test has the smallest M. A. E.? [Same as part (b) of the question in DeGroot and Schervish, except the number 1 / 3 is replaced by the variable y. Of course, plugging in y = 1 / 3 gives the answer in the back of the book.]

8. A brand of raisin bran averages 84.2 raisins per box. The boxes are filled from large bins of well mixed raisin bran. What is the standard deviation of the number of raisins per box?

9. Let X be the number of winners of a lottery. If we assume that players pick their lottery numbers at random, then their choices are i. i. d. random variables and X is binomially distributed. Since the mean number of winners is small, the Poisson approximation is very good. Hence we may assume that X has a Poisson distribution with mean n p, where n is the total number of tickets sold and p is the probability of any particular ticket winning.

Because of our independence assumption, what other players do is independent of what you do. Hence the conditional distribution of the number of other winners given that you win is also Poisson with mean n p. If you are lucky enough to win, you must split the prize with X other winners. You win A / (X + 1) where A is the total prize money. Thus

E{A / (X + 1)}
is your expected winnings given that you win. Calculate this expectation (as a function of the constants A, n, and p).

## Review Problems

1. The function

f(x) = (60 / 11) (x - 2 x2 + 3 x3 - 2 x4),     0 < x < 1
is the probability density function of a random variable X.
1. Calculate E(X).
2. Calculate var(X).

Answers. (a) 6 / 11. (b) 45 / 847. Not asked but a useful auxiliary result: E(X2) = 27 / 77.

2. Suppose var(X) = 3.

1. Calculate var(4 X + 5).
2. Calculate sd(4 X + 5).

Answers. (a) 48. (b) 4 √3.

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