University of Minnesota, Twin Cities School of Statistics Stat 5101 Rweb
General Information Homework Assignments Homework Solutions Exam Solutions Material Covered
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 gof 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 calculatedin 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
Find the cumulative distribution function of X.
Be sure to define your answer on the whole real line.
2. Suppose the cumulative distribution function F of a random variable X is defined by
Find the probability density function of X.
Be sure to define your answer on the whole real line.
3. Suppose the joint probability density function f of random variables X and Y is defined by
4. Suppose X has probability density
5. Suppose X has probability density
6. Suppose X, Y, and Z are random variables such that
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
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
1. The function
Answers. (a) 6 / 11. (b) 45 / 847. Not asked but a useful auxiliary result: E(X^{2}) = 27 / 77.
2. Suppose var(X) = 3.
Answers. (a) 48. (b) 4 √3.
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