# Homework Assignments for Stat 515H – Fall 2015

Here are three comments concerning the 6th, 7th, 8th, and 9th editions of the textbook.
1. The main difference in the assigned problems from the 6th, 7th, 8th, and 9th editions is in the page numbers. In the list of assignments, the notation pWWW / pXXX / pYYY / pgZZZ refers to page WWW in the 6th edition, page XXX in the 7th edition, page YYY in the 8th edition, and page ZZZ in the 9th edition. Other differences are noted similarly.
2. In assignments 2 and 3 the problem numbers are the same in all three editions and are given only once.
3. The pdf files of the assignments are photocopied from the 6th edition.

• Assignment 1.   Due Friday, September 18, 2015. A pdf file of Assignment 1 is available online. This assignment concerns the birthday problem, which was discussed in class, and a related problem that I call the hotel coincidence problem. Relevant formulas are summarized on page 1 of the attached file. The 4 questions to answer are given on page 2; the first 3 are related to the birthday problem, and the fourth relates to the hotel coincidence problem. Hints for solving the hotel coincidence problem are available online.

• Assignment 2.   Due Friday, September 25, 2015. A pdf file of Assignment 2 is available online. The following problems concern the material in Chapter 1.
1. p16 / p16 / p16 / pg15, #5.
2. p16 / p16 / p16 / pg15, #8 (a)–(d).
3. p16 / p16 / p16 / pg15, #10 (a), (b), (d), (e).
4. p16 / p17 / p16 / pg16, #12 (a), (b).
5. p18 / p18 / p17 / pg16, #25. This problem involves multinomial coefficients.

• Assignment 3.   Due Wednesday, September 30, 2015. A pdf file of Assignment 3 is available online. The following problems concern the material in Chapter 2. Do all the parts of the problems unless otherwise stated.
1. p53 / p55 / p50 / pg48, #3. In place of the last sentence of the problem as stated in the book, list all the outcomes in EF and in FG, describe E∪F and EF-complement in words, and show that EFG=FG.
2. p54 / p56 / p50 / pg48, #6 (a)–(d).
3. p54 / p56 / p51 / pg48, #8 (a)–(c).
4. p59 / p61 / p54 / pg52, #1.
5. p59 / p61 / p54 / pg52, #2.
6. p59 / p61 / p54 / pg52, #3.
7. p59 / p61 / p55 / pg52, #6 (a)–(j).
8. p60 / p62 / p55 / pg53, #12.
9. p60 / p62 / p55 / pg53, #13.

• Assignment 4.   Due Monday, October 5, 2015. A pdf file of Assignment 4 is available online. The following problems concern the material in Chapter 3. Pass in only the first 6 problems. Do not pass in problems 7, 8, and 9.
1. p104 / p111, #1 / p102 / pg97, #3.1.
2. p104 / p111, #2 / p102 / pg97, #3.2.
3. p104 / p111, #4 / p102 / pg97, #3.4
4. p104, #11 / p112, #15 / p102 / pg98, #3.15.
5. p104, #12 / p112, #16 / p102 / pg98, #3.16.
6. p104, #13 / p112–113, #17 (a), (b) / p102 / pg98, #3.17 (a), (b).
7. p115 / p124, #1 / p110 / pg106, #3.1.
8. p115 / p124, #2 / p110 / pg106, #3.2.
9. A final problem is a simpler version of the exercise on p116 / p124, #6 / p111 / pg107, #3.6.
For the statements of all the problems, including this final problem, and for hints on p104, #11 / p112, #15 / p102 / pg98, #3.15 and on p104, #12 / p112, #16 / p102 / pg98, #3.16, consult the pdf file of Assignment 4.

• Assignment 5.   Due Monday, October 19, 2015. A pdf file of Assignment 5 is available online. The following problems concern the material in Chapter 4.
1. p171 / p187, #2 / p172 / pg163, #4.2.
2. p171 / p187, #7 (a)–(d) / p173 / pg163, #4.7 (a)–(d).
3. p171 / p187, #8 (a), (d) / p173 / pg163, #4.8 (a), (d).
4. p173 / p190, first 3 sentences of #22 / p174 / pg164, first three sentences of #4.22. In the third sentence solve only part (a) i=2. Do not answer the question in the last sentence of the problem.
5. p180 / p197, #5 / p180 / pg169, #4.3. Do only the case α > 0. You are given that F(x) = P(X \leq x) for all real x. By a simple change of variables express P(α X + β \leq x) in terms of F(x).
6. p180 / p198, #8 / p180 / pg170, #4.6.
7. p180 / p198, #9 / p180 / pg170, #4.7.

• Assignment 6.   Due Wednesday, October 30, 2015. A pdf file of Assignment 6 is available online. Hints for the second problem are also available. The following problems concern the material in Chapter 4.
1. p173 / p189, #19 / p174 / pg164, #4.19.
2. p173 / p189–190, #20 (a), (c) / p174 / pg164, #4.20 (a), (c). A hint for (c) is that the possible values of X are 1, –1, –3.
3. p173 / p190, #21 (b) / p174 / pg164, #4.21 (b).
4. p175 / p192, #35 (a), (b) / p176 / pg166, #4.35 (a), (b). A hint is to first calculate P(X=1.10) and P(X=–1.00).
5. p176 / p192, #38 (a), (b) / p176 / pg166, #4.38 (a), (b).
6. p177 / p194, #51 (a), (b) / p177 / pg167, #4.51 (a), (b). A hint is to use a Poisson distribution and leave answers in terms of the real number e.
7. p177 / p194, #57 (a), (b) / p177 / pg167, #4.57, (a), (b). Leave answers in terms of the real number e.
8. p177 / p194, #58 (a), (b) / p177 / pg167, #4.58 (a), (b).

• Assignment 7.   Due Friday, November 13, 2015. In order to answer 3 parts of p229 / p249, #15 / p225 / pg213, #5.15 the values of the normal distribution must be interpolated from Table 5.1. The method is discussed in the handout Interpolating Φ(x) for 0 < x < 3.49 Not in Table 5.1. The following problems concern the material in Chapter 5. A pdf file of Assignment 7 is available online.
1. p228 / p247, #1 (a), (b) / p224 / pg212, #5.1 (a), (b).
2. p228 / p247, #4 (a)–(c) / p224 / pg212, #5.4 (a)–(c). In order to answer (c), assume the independence of the events that the devices exceed 15 hours. Leave your answer to (c) as a sum without simplifying it.
3. p228 / p248, #6 (a)–(c) / p224 / pg212, #5.6 (a)–(c). A hint for (a) is to integrate by parts twice.
4. p229 / p248, #13 (a), (b) / p225 / pg212, #5.13 (a), (b).
5. p229 / p248, first question in #14 / p225 / pg213, first question in #5.14. Compute E[Xn] by using Proposition 2.1 on page 192 / page 210 / page 191 / page 181. Do not check the result by using the definition of expectation.
6. p229 / p249, #15 (a)–(e) / p225 / pg213, #5.15 (a)–(e). In your solution, indicate how you transform the given probabilities into probabilities involving a standard normal random variable and then use Table 5.1 on page 203 / page 222 / page 201 / page 190 to get numerical answers. Use Exercise 4b on pages 203–204 / pages 221–223 / page 202 / page 191 as a model. The answers to parts (a), (b), and (c) given in the back of the textbook are incorrect; they should be .7976, .6826, and .3694. The answers to parts (d) and (e) given in the back of the textbook are correct (.9522 and .1587).

• Assignment 8.   Due Monday, November 23, 2015. The following problems concern the material in Chapter 5. A pdf file of Assignment 8 is available online.
1. p229 / p248, #11 / p225 / pg212, #5.11. Let X be a uniform random variable on the interval [0,L]. A hint is that the ratio of the shorter to the longer segment is min{X/(L–X), (L–X)/X}.
2. p229 / p249, #16 / p225 / pg213, #5.16. Assume that the annual rainfalls from year to year are independent. To solve this problem, find P{X £ 50} from Table 5.1 on page 203 / page 222 / page 201 / page 190. Do not simplify the final answer.
3. p230 / p249, #18 / p225 / pg213, #5.18. Use Table 5.1 on page 203 / page 222 / page 201 / page 190.
4. p230 / p249, #19 / p225 / pg213, #5.19. Use Table 5.1 on page 203 / page 222 / page 201 / page 190. The answer in the book makes no sense. The correct answer is c = 14.56.
5. p230 / p249, #25 / p226 / pg213, #5.25. Here is a hint for this problem. Let X be a binomial random variables with n=150 and p=.05. The probability in question is P(0 £ X £ 10) = P(–.5 £ X £ 10.5). Then use the normal approximation, which is discussed in Section 5.4.1, and use Table 5.1 on page 203 / page 222 / page 201 / page 190. When you calculate the approximate probability involving the standard normal random variable, round off the endpoints of the interval to two decimal places, allowing you to avoid interpolating the values.
6. p231 / p250, #32(a), (b) / p226 / pg214, #5.32 (a), (b).
7. p231 / p250, #34 / p226 / pg214, #5.34.

• Assignment 9.  Due Wednesday, December 9, 2015. This is the last assignment. Out of consideration for the grader, December 9 is the last day on which this homework and any late homeworks will be accepted. The following problems concern the material in Chapter 6. A pdf file of Assignment 9 is available online.
1. p290 / p313, #1 (b) / p287 / pg271, #6.1 (b). In order to answer this, determine the value of X and Y for each of the 36 possible outcomes. A table might help you organize the calculation.
2. pp290–291 / p314, #9 (a)–(c) / p287 / pg271, #6.9 (a)–(c). In order to answer part (c), integrate the joint probability density function over the set {(x,y) : 0 < x < 1, 0 < y < x}. First do the y-integration and then do the x-integration.
3. p291 / p314, #10 (a), (b) / p287 / pg271, #6.10 (a), (b). Also determine the (marginal) probability density function of X and the (marginal) probability density function of Y, using the formulas in the two displays before Example 1c on page 243 / page 262 / page 236 / page 224.
4. p292 / p315, #20 / p288 / pg272, #6.20. Do only the first example, using Proposition 2.1 on page 253 / page 272 / page 245 / page 233. Also determine the (marginal) probability density function of X and the (marginal) probability density function of Y.
5. p292 / p316, #23 (a)–(e) / p288 / pg272, #6.23 (a)–(e). Before answering parts (b)–(e), determine the (marginal) probability density function of X and the (marginal) probability density function of Y. Then use this information to answer parts (b)–(e).
6. p293 / p317, #33 (b) / p289 / pg273, #6.30 (b). Show all the steps leading to the answer, which is approximately 0.2119. In order to solve this problem, use Proposition 3.2 on page 264 / page 283 / page 256 / page 243 to express the sum X+Y as an appropriate normal random variable. Use part (a) of Example 3c on page 265 / pages 284–285 / pages 257–258 / pages 244–245 as a model but without the continuity correction, which should not be introduced here. The continuity correction should not be introduced because the inequality in the desired probability is strict.