M362K, Smith, Sp 98

ASSIGNMENT FOR THURSDAY, JANUARY 22

 1. Course Information
 2. Hints on Textbook Reading
 3. Guidelines for Written Homework

Bring any questions you have on these to class.

II. Look over the handout Some Things You Need to Know from Calculus and Precalculus (First Day Packet plus additional half sheet). If you need review, start doing it now.

III. Read Sections 1.1, 2.1, and 2.2 of the textbook. The first two are easy reading . The third will take more time. As you  read section 2.2, be sure to think about the questions in parentheses. I will probably ask you about them in class.
Note for non-biologists regarding Example 2.2.6: (Biologists, please correct  me if I’m wrong) Alleles are, roughly speaking, different forms  of a gene that can  occur at a gene site. In popular usage, they are often erroneously referred  to as genes.

IV. Answer the following Questions as you come to them in your reading: 2.2.1, 2.2.2, 2.2.6, 2.2.8, 2.2.12, 2.2.20, 2.2.21, 2.2.24, 2.2.27. Most of these have answers or at least hints in the back of the book. Try the question on your own and don’t consult the back of the book until you have your answer or get stuck.
 

Optional: Do one or more of the following and bring your data to class:

 1. Toss a single thumbtack 100 or more times and treat the data as we did the thumbtack data in class: Record  the number of  point down landings for each group of  ten tosses,  then calculate the sequence of proportion of point-down for ten, twenty, thirty, etc. Also make a histogram of the proportions out of each group of ten.

 2. Instead of a thumbtack, use a coin and count heads.

 3. Use a penny, but instead of tossing it, spin it: Place your penny on its edge, with Lincoln right side up and facing you. Hold the penny lightly with one finger of one hand. Flick the edge of the penny sharply with a finger of your other hand to set it spinning. (This may take  a little practice if you’re a novice.) Let the penny spin freely until it falls. If it hits something while spinning, don’t count that trial. (I am told this is especially interesting for pennies minted around 1962, so if you have one of that vintage you might try this option.)

 4. For the steady of hand: Instead of tossing or spinning the penny, set it on edge, then slam your hand down on the table so that the penny falls on one side or the other. (This only works well with fairly new pennies.)

 5. Create and run a coin tossing simulation on a computer or calculator. You might try a weighted coin -- say where the probability of heads is 40%. If you do this one, do at least 1000 tries.
 

ASSIGNMENT FOR TUESDAY, JANUARY 27

II. Do Questions 2.3.2, 2.3.3, 2.3.4, and 2.3.6

III. Read  section 2.4 through the top of p. 31
Comments:
 The first paragraph  is awkwardly written,  so may be unclear.  What  it is trying to say is: If a sample space S is either finite or countably infinite, then any function P on the sample space which satisfies conditions (1) and (2) allows us to define a probability function on S by P(A) = (the summation given in the text). Such a probability function is called a discrete probability function.
 Be sure to answer  the parenthetical questions on pp. 27 and 29.

IV: Do Questions 2.4.3, 2.4.4, 2.4.8, 2.4.9

Optional:
 Read Example 2.4.5
 Do Question 2.4.1
 
 
 

ASSIGNMENT FOR THURSDAY,  JANUARY 29

II. Do questions 2.5.1(a)  and (b),  2.5.2, 2.5.10, and 2.5.14 as you come to them in the reading.

III. Turn in: Solutions to Questions 2.2.28, 2.3.1, 2.3.5, 2.3.7 (Review  the Guidelines for Writing Homework as you do this.)

Optional: Question 2.5.1(c)
 
 

Please note: Subscripts and superscripts appear at "ground level" in the following.

ASSIGNMENT FOR TUESDAY, FEBRUARY 3

I. Read Section 2.6, omitting Example 2.6.8 (p. 57). Be sure to think about the parenthetical questions on pp. 53 (Example 2.6.2), 55 (example 2.6.5 -- it’s OK just to do the case of K3 for this one), 56 (Example 2.6.6),  57 (Example 2.6.7 -- draw a Venn diagram), 59 (proof of Theorem 2.6.1), and 61 (Example 2.6.11).

Comment: Theorem 2.6.1 may look complicated at first, but if you notice the pattern, it becomes simpler to remember  and understand. Notice also the important special case: P(B) = P(B|A) + P(B|AC)

II. Do the following exercises as you come to them in your reading: 2.6.2, 2.6.8, 2.6.9, 2.6.10, 2.6.21, 2.6.23.

ASSIGNMENT FOR THURSDAY, FEBRUARY 5

I. Read Section 2.7.
Comments:
 1. The formula in Bayes’ Theorem may look complicated at first glance, but notice from the proof that it is just a matter of putting together the definition of conditional probability (Definition 2.6.1) and Theorem 2.6.1 (See comment above.)  As  with Theorem 2.6.1, note the special case when n = 2,  A1 = A  and A2 = AC.
 2. The explanation of Example 2.7.1 is a little confusing. You may be able to explain it better yourself.
 3. Be sure to think about the parenthetical questions on pp. 65 (Example 2.7.2) and 66 (Example 2.7.3 -- there are two ways to do this ; which is better?).
 4. Note that you can do part (b) of Example 2.7.3 a shorter way by using the result of part (a).
 5. In class, we will look at Example 2.7.4 from the relative frequency view of probability.

II. Do the following exercises:  2.7.1, 2.7.3
Hints: Start the exercises by giving names to the events involved (e.g., M: has mono, S: has sore throat), then write out the probabilities and conditional probabilities given in the problem. Now write out what conditional probability you need to find, and apply Bayes’ Theorem to find it. You will only need the special case of n = 2, so Example 2.7.3 is a good model to follow.

III. Turn in: Solutions to Exercises 2.4.5, 2.5.3, 2.5.16, 2.6.14

Note: First exam will be Thursday,February 19, usual class time, in Burdine 212
 
 
 
ASSIGNMENT FOR TUESDAY, FEBRUARY 10

I. Read section 2.8 plus pp. 80 to the bottom of p. 81. Be sure to think about the parenthetical questions in Examples 2.8.2, 2.8.7, 2.8.8, and 2.8.9. You may need to fill in some details in Examples 2.8.2, 2.8.3, and 2.8.6.

II. Do the following exercises as you come to them in your reading: 2.8.2, 2.8.3, 2.8.15, 2.8.16, 1.9.1.

Optional: Exercise 2.2.18

ASSIGNMENT FOR THURSDAY,  FEBRUARY 12

I. Read the rest of Section 2.9. (You may omit Examples 2.9.3 and 2.9.5.)

II. Do the following exercises to discuss: 2.8.22, 2.9.4 (Note: “wired in series” means that all the bulbs must work for the lights to be on.), plus the following:

1. Corrected version Players A, and B toss a fair coin. The first to throw a head wins. What are their respective chances of winning? (Note: This is a simpler version of Question 2.9.5. What was originally stated on the assignment sheet handed out is exactly Question 2.9.5.)

2. In a certain lottery game, your chances of winning are 1/60. How many times would you have to play to have probability at least 1/4 of winning at least once?

3. According to a 1995 article in Newsweek, the chance of HIV transmission from female to male in unprotected sex is about 1 in 400. What would be the probability that a man having unprotected sex with 20 different women would become infected with HIV?

Optional: Exercise 2.9.5

III. Hand in: Exercises 2.6.26, 2.7.2, 2.7.5, 2.8.6

ASSIGNMENT FOR TUESDAY, FEBRUARY 17

I. Read Section 3.1 plus pp. 96 - 113, omitting Case Study 3.1 and Examples 3.2.10 and 3.2.11.

Comments:  On p. 100, the equal sign with a dot over  it means “approximately equal to.”
 The case  r = 2 in the expression in Theorem 3.2.3 is the binomial coefficient  “n choose k”, where k = n1. This will be discussed  more in the next section.

Optional: Try out Stirling’s Formula (p. 100) for n = 2 through 6 (or higher, if you wish).

II. Do the following exercises as you come to them in your reading: 3.2.1, 3.2.4, 3.2.10, 3.2.11, 3.2.20, 3.2.23

 
 THURSDAY, FEBRUARY 19

 Exam in Burdine 212, usual class time. Covers  through Section 2.9. Problems will be roughly the same level of difficulty as homework problems. Be sure you know the basic definitions, facts and formulas, and how to use them. Since deciding what technique or concept to use when is an important part of taking the exam, here are some problems to practice on. Caveat: These problems are not intended as a guide to what will or will not be on the exam.

Warning: Symbols such as intersection and union  signs  do not appear on this web version. Subscripts and superscripts are on the same level as regular text.

1. Which of the following is more probable?
 a. The world will  come to an end in the next 50 years.
 b.  The earth will be destroyed  by either a meteorite or a nuclear war within the next  50 years.

(Comment: On the survey  conducted the first class day,  7 people assigned a higher probability to (a) than (b), 12 assigned equal probabilities, 16 assigned (a) a smaller probability than (b), and 3 left the question blank.)

2. In six repeated tosses of a fair coin,  which event , if either, is more likely, HHTHHT or HHHHHH? (On the first class day survey, 12 out of 37 said the first option is more likely; the rest said the events  are equally likely. In a survey of 39 students, mainly freshmen, last semester,  22 out of 39 said the first option is more likely; the rest said the events  are equally likely.)
 

3. Guess to test your intuition before you calculate on this one. (Note: I botched this question on the first day survey -- I gave the wrong definitions of “false positive” and “false negative”, which made the problem much easier than it is.) Suppose 1 in 1000 people have a disease. A test for it has a 10% false positive rate (that is, 10% of people without  the disease test positive) and a 10% false negative rate (that is, 10% of the people with the disease test  negative).  If someone tests positive for the disease, what are the chances that they actually have it?
 How would the answer change if only 1 in 10,000 had the disease? Or if 1 in 100 had it?

4. Suppose you routinely check coin-return slots in vending machines to see if they have any money in them. You have found that about 10% of the time you find money.
 a. What is the probability that you do not find money the next time you check?
 b. What is the probability that the next time you will find money is on the third try?
 c. What is the probability that you will have found money by the third try?
 d. What  assumption are you making in your solutions? Why do you think this assumption is warranted?

5. Suppose that A and B are mutually exclusive events for which P(A) = .3 and P(B) = .5. What  is the probability that
 a. either A or B occurs
 b. A occurs but B does not
 c. both A and B occur?

6. Two fair dice are rolled. What is the probability that at least one lands on 6 given that the dice land on different numbers?

7. A customer  visiting the suit department of a certain store will purchase a suit with probability .22, a shirt with probability .30, and a tie with probability .28. The customer will purchase both a suit and a shirt with probability .11, both a suit and a tie with probability .14, and both a shirt and a tie with probability .10. A customer will purchase all 3 items with probability .06. What  is the probability that a customer purchases
 a. none of theses  items
 b. exactly one of these items?

8.  Two dice have each had two of their sides painted red, two painted black, one painted yellow, and the other painted white. When the pair of dice is tossed, what is the probability that both land on the same color?

9. Note: Printed version has a misprint -- the exponent 2 is outside the parentheses, not inside.A filling station is supplied with gasoline once a week. Its weekly volume of sales in thousands of gallons is described by the probability function (aka probability density function)

                    c(1 - x)2,  0 < x < 1
  f(x) =
                     0      otherwise.

 a. What  is the value of c?
 b. What does the capacity of the tank need to be so that the probability of the supply’s being exhausted in a given week is .01?

10. Let A and B be events having positive probability. For each of the following statements, decide whether the statement  is (i) necessarily true, (ii), necessarily false, or (iii) possibly but not necessarily true. Give explanations.
 a. If A and B are mutually exclusive, then they are independent.
 b. If A and B are independent, then they are mutually exclusive.
 c. P(A) = P(B) = .6 and A and B are mutually exclusive.
 d. P(A) = P(B) = .6 and A and B are independent.

11. If a die is rolled 4 times, what is the probability that 6 comes up at least once?

12. An ectopic pregnancy is twice as likely to develop when the pregnant woman is a smoker as it is when she is a nonsmoker. If 32 percent of women of childbearing age are smokers, what percentage of women having ectopic pregnancies are smokers?

13. Let E, F, and G be three events. Find expressions for the events so that of E, F, and G:
 a. only E occurs
 b. both E and G but not F occurs
 c. at least one of the events occurs
 d. at least two of the events occur
 e. all three occur
 f.  none of the events occurs
 g. at most  one of  them occurs
 h. at most  two of them occur
 I. exactly two of them occur
 j. at most  three of them occur.
 

14. Find a formula in terms of P(A), P(B), and P(A  B)for the probability that exactly one of the events  A or B occurs.
 

15. Correction to printed version: The function is 10 over x-squared for x > 10, not  10 over x.The probability function (aka probability density function)  describing the lifetime (measured in hours) of a certain type of electronic device  is given by

                10/x2  x  > 10
  f(x)
                0  x  < 10.

Find the probability that the device  lasts more than 20 hours.

16. According to one source, the probability of being injured by lightning in any given year is 1/685,000.
 a. What is the probability that someone  who lives 70 years will never be injured by lightning? (What  assumption are you making, and why is it reasonable?)
 b. Do the probabilities in this exercise apply specifically to you? Why or why not?
 c. The population of the U.S. is about  60 million. In a typical year,  about how many people in the U.S. would be expected  to be injured by lightning?

17. In a certain community, 36 percent of the families own a dog, and 22 percent of the families that own a dog also own a cat. In addition, 30 percent of the families own a cat. What is
 a. the probability that a randomly selected family owns both a dog and a cat;
 b. the probability that a randomly selected  cat-owning family also owns a dog?

18. A couple has two children. What is the probability that both are girls if the eldest is a girl?
 
 
 

ASSIGNMENT FOR TUESDAY, FEBRUARY 24

I. Read Section 3.3 through p. 117 only, Section 3.5 through the middle of p. 132 only, and Section 3.6 through the top of p. 150. Be sure to think about the parenthetical question at the top of p. 149.

II. Do the following Questions as you come to them in your reading:  3.3.1, 3.3.4, 3.5.1, 3.5.6, 3.6.1, 3.6.4, 3.6.21, 3.6.24.
 
 

ASSIGNMENT FOR THURSDAY, FEBRUARY 26

I. Read pp. 152 - top of p. 154 and Examples 3.6.8 - 3.6.12.

II. Do the following Questions as you come to them in your reading: 3.6.25, 3.6.26, 3.6.33

III. Do the following exercises to hand in. Be sure to explain your reasoning!

1. Texas license plates issued during a certain period of years have four letters (in places 1, 2, 3, and 6) and two numbers (in places 4 and 5).
 a. How many different license plates of this type are possible? (Letters and numbers may be repeated.)
 b. How many license plates of this type are possible with the additional restriction that no letter or number may be repeated?
 c. How many plates of this type (again allowing repetitions, as in part (a)) are possible if the numbers 0 and 1 and the letters I and O are not allowed? (Some states sometimes exclude these to avoid possible confusion of 0 with O and I with 1.)
 d. How many plates would be possible if the only restriction were that there had to be four letters and 2 numbers, but the letters and numbers could be in any places?

2. How many eleven letter arrangements of letters are possible from the letters MISSISSIPPI?

3. In the game of bridge, the entire deck of 52 cards is dealt out to 4 players.
 a. How many ways  is it possible to do this?
 b. In how many of these does the first player have all the hearts?
 c. What  is the probability that the first player will be dealt a hand of all hearts?

ASSIGNMENT FOR TUESDAY, MARCH 3

I. Read  pp. 169 - 178, Definition 4.2.2 (p. 181), Example 4.27 (p. 183) and pp. 184 - 185. (You may ignore the Comment on p. 170.) Be sure to think about  the parenthetical comments on p. 170 (Example 4.1.2 -- do the second question just for the case of two tosses rather than four.),  p. 171 (Examples 4.1.3 and 4.1.4), p. 173 (Example 4.2.2), p. 174 Example 4.2.3), and p. 176 (Example 4.2.5).

II. Do the following Questions as you come to them in your reading: 4.2.5, 4.2.7, 4.2.11, 4.2.14, 4.2.19, 4.2.25, 4.2.27
 
 

ASSIGNMENT FOR THURSDAY, MARCH 5

I. Read  Section 4.3 (You may  ignore Theorems 4.3.1 and 4.3.3.) Think about the parenthetical comment on p. 194 (Example 4.3.2) Fill in the details at the end of Example 4.3.3. Try t he parenthetical comment at the end of Example 4.3.3.

II. Do the following Questions as you come to them in your reading: 4.3.1, 4.3.7(a -- but think about  how you would do b ), 4.3.8, 4.3.10, 4.3.13

III. Turn in solutions to the following problems:

1. A coin is biased so that heads is three times as likely as tails. Let the random variable X be the total number of heads in three independent tosses of the coin. Find
 a. The probability density function of X.
 b. The cumulative distribution function of X.

2. A  purchaser of electrical components buys them in lots of size 10. He randomly selects three components from each lot to inspect and accepts  the lot only if all 3 are nondefective. If 30 percent of the lots have 4 defective components and 70 percent have only one defective component, what proportion of lots does the purchaser reject?

(Caution:  In the following problems, you must figure out where subscripts, superscripts, and missing symbols go.)
3. a. Find the cumulative distribution function of the random variable X whose probability density function is

                      1/3     for 0 < x < 1
  fX(x) =      1/3      for 2 < x < 4
                       0        elsewhere

 b. Sketch the graphs of the probability density and cumulative distribution functions. (Indicate clearly which is which.)

4. The random variable X has cumulative distribution function

                       1 - (1 + x) ex       for x > 0  CORRECTION 3/4/98: e to the exponent -x, not x
 FX(x) =
                       0                              for x < 0

Find
 a. P(1<X<3)
 b. P(X > 4)
 c. The probability density function of X.

ASSIGNMENT FOR  TUESDAY, MARCH 10

I. Do the following questions from Section 4.3: 4.3.25, 4.3.30

II. Read Section 4.4 through the bottom of p. 207. The proof of the continuous case of Theorem 4.4.1 is optional. Do the parenthetically requested verification in Example 4.4.1 (p. 207).
 Also read Section 4.5. (Example 4.5.13 optional.) Do the case a < 0 in the proof of Theorem 4.5.1 and the special case of the parenthetical question in Example 4.5.6 (p.  217) where X is the number appearing on one die and Y is the number appearing on another die.

III. Do the following Questions as you come to them in your reading: 4.4.1 - 4.4.3, 4.5.1, 4.5.6, 4.5.16
 
 

ASSIGNMENT FOR THURSDAY, MARCH 12

I. Read Section 4.6

II. Do the following questions as you come to them: 4.6.3, 4.6.4, 4.6.9, 4.6.11(c)

III. Turn in: Questions 4.2.28, 4.3.14, 4.4.4, 4.5.5
 
 

NEXT EXAM: Thursday, April 2. Will cover  (what we have covered  from) Chapters 3 and 4, plus possibly the first part of Chapter 5. (More details later.)

ASSIGNMENT FOR  TUESDAY, MARCH 24

I. Read Sections 5.1 and 5.2 (Example 5.2.8 and the comment following it are optional.) Answer the parenthetical questions  in the following examples: 5.2.1 (p. 231), 5.2.2 (p. 232), 5.2.3 (p. 233), 5.2.4 (p. 234; also fill in any missing details in this example), 5.2.6 (if you have a graphing calculator or suitable computer software), and 5.2.7 (p. 239. Approach the second question intuitively.)

II. Do the following Questions as you come to them in your reading: 5.2.2, 5.2.6 (Hint: See Example 5.2.3), 5.2.13 (hint: See Example 5.2.5), 5.2.18 a, b, d, e.

ASSIGNMENT FOR THURSDAY, MARCH 26

I. Read section 5.3, omitting pp. 247 - 253, Example 5.3.9, and Examples 5.3.11 - 5.3.13. (Examples 5.3.3 and  5.3.4 are  optional.) Answer the parenthetical question in Example 5.3.10 (p. 258).

II. Do the following questions as you come to them in your reading: 5.3.3, 5.3.4, 5.3.5 (Hint: Identify the random variable whose expectation you must find and write it as a linear combination of two simpler random variables whose expectations you can find by the result of a previous example. To get the answer in the back of the book , you must assume p + q = 1.), 5.3.14, and 5.20.

III. Turn in: Questions 4.6.2, 4.6.5, 4.6.12, plus the following problem:

ASSIGNMENT FOR  TUESDAY, MARCH 24

I. Read Sections 5.1 and 5.2 (Example 5.2.8 and the comment following it are optional.) Answer the parenthetical questions  in the following examples: 5.2.1 (p. 231), 5.2.2 (p. 232), 5.2.3 (p. 233), 5.2.4 (p. 234; also fill in any missing details in this example), 5.2.6 (if you have a graphing calculator or suitable computer software), and 5.2.7 (p. 239. Approach the second question intuitively.)

II. Do the following Questions as you come to them in your reading: 5.2.2, 5.2.6 (Hint: See Example 5.2.3), 5.2.13 (hint: See Example 5.2.5), 5.2.18 a, b, d, e.

ASSIGNMENT FOR THURSDAY, MARCH 26

I. Read section 5.3, omitting pp. 247 - 253, Example 5.3.9, and Examples 5.3.11 - 5.3.13. (Examples 5.3.3 and  5.3.4 are  optional.) Answer the parenthetical question in Example 5.3.10 (p. 258).

II. Do the following questions as you come to them in your reading: 5.3.3, 5.3.4, 5.3.5 (Hint: Identify the random variable whose expectation you must find and write it as a linear combination of two simpler random variables whose expectations you can find by the result of a previous example. To get the answer in the back of the book , you must assume p + q = 1.), 5.3.14, and 5.20.

III. Turn in: Questions 4.6.2, 4.6.5, 4.6.12, plus the following problem:

 Certain coded measurements of the pitch diameter  of threads of a fitting have the probability density function  f(x) = :

     4/(?(1 + x^2) )     for  0 < x < 1                    (If you see a ?, read pi)
 

     0      elsewhere

Find the expected value of this random variable.
 

Note: Exam Thursday, April 2. Will cover  (what  we have covered of) Chapters 3 and 4, plus sections 5.1 and 5.2, and of course previous material as needed. (Item in italics is a correction to the original.)
 

ASSIGNMENT FOR  TUESDAY, MARCH 31

I. Read Section 5.4, omitting pp. 267 - 268 and 275 (bottom) - 278. Prove Theorem 5.44 (p. 273). Go through the proof of Theorem 5.4.5 in the case of two random variables.

II. Do the following questions as you come to them in your reading: 5.4.1, 5.4,3, 5.4.5, 5.4.13, 5.4.14, 5.4.16, 5.4.19, 5.4.25

III.  Find the variance for  each of the following random variables:

 1. X = the number on a fair die that is tossed once.

 2. Y =  the sum of the numbers obtained by tossing a fair die 100 times. [Hint: Let Xi be the number that comes up on the ith toss; use Theorem 5.4.3.]

 3. Z =  the number of heads in n tosses of a fair coin. [Hint: Let Xi  be 1 if the ith toss is heads, 0 if the ith toss is tails; use Theorem 5.4.3.]

THURSDAY, APRIL 2

Exam. Will cover  (what  we have covered of) Chapters 3 and 4, plus sections 5.1 and 5.2, and of course previous material as needed. Room: Burdine 212. Note that this is changed from the earlier announced room. See  Review for Second Exam

ASSIGNMENT FOR  TUESDAY, APRIL 7

I. Review what we have not yet  discussed of the assignment for Tuesday, March 31:
 A. pp. 272 - top of p.275 (including the proof of Theorem 5.4.4, and the proof for the case of two random variables of Theorem 5.4.5)

 B. Questions 5.4.19, 5.4.25, and Problems III (2) and III (3) of the assignment for March 31.

II. Read pp. 279 - 280 and 286 - top of 288.

III. Do the first part of Question 5.5.1 (assume a and b are > 0 ) and Question 5.5.12
 

ASSIGNMENT FOR THURSDAY, APRIL 9

I. Read from the bottom of p. 288 to the top of p. 292. Caution: There are some misprints; in particular, some x’s appear as X’s. BE sure to think about the parenthetical question at the top of p. 290.

II. Do the following questions as you come to them in your reading: 5.5.17, 5.5.21, 5.5.22 (Hint: Example 5.5.6 and Theorem 5.5.4(b)), plus:

 1. The random variable X has moment generating function

M(t) = (1/4 et + 3/4)20. What  is P{X = 0}? (Hint: Example 5.5.5 and Theorem 5.5.3)

III. Turn In: Note: There was a misprint in the original version of the assignment;  Question 5.3.8 is to  be handed in, not Question 5.4.8.
Questions 5.3.8 (Hint: Example 5.3.1 can shorten your work a little.), 5.3.11 (Hint: Write the random variable in question as a sum of random variables whose expected values are fairly easy to find.), 5.4.17, 5.4.23

ASSIGNMENT FOR  TUESDAY, APRIL 14

I. Read Section 5.6. (The Comments on p. 294 and p. 295 are optional.) Be especially sure to read slowly, for understanding.

II. Do the following Questions: 5.6.1 (Hints: Use Problem III (2) from March 31. Once you get µ, the problem will make more sense.), 5.6.3 (Hint: Use Problem III(3) from March 31.), 5.6.7
 

ASSIGNMENT FOR THURSDAY, APRIL 16

I. Read Sections 6.1 and 6.2.

II. Do the following Questions as you come to them in your reading: 6.2.2 (Try it two ways.), 6.2.4, 6.2.8, 6.2.9 (Note: Question 6.2.7 is the same as Question 5.5.18, assigned to be handed in today.)

III. Turn in: Questions 5.5.18, 5.6.2, 5.6.4 (a), plus:

1. The correlation  r(X,Y) of  two random variables X and Y is defined by

 r(X,Y) = Cov(X,Y)[Var(x) Var(Y)]  -1/2  .  [The -1/2 is an exponent.]

Alternatively,

 r(X,Y) = Cov(X,Y)/sXsY    ,

where sX and sY  are the standard deviations of X and Y, respectively. [These should be "sigma sub X", etc.]

a. Expand Var(X/ sX + Y/ sY) and use the definition of r(X, Y) to show that
r(X,Y)  * -1.

 
b. Proceed similarly with Var(X/ sX - Y/ sY) to show that  r(X,Y)  ? 1.

c.  Use the properties of Var  and Cov  to  calculate r(X,Y) when X = aY + b. (Be careful to distinguish between the cases  a > 0 and a < 0.)
 
 

ASSIGNMENT FOR  TUESDAY, APRIL 21

I. Do Questions 6.2.13 and 6.2.16

II. Read Section 6.3 (Note: There is a misprint in the statement of Definition 6.3.1 -- the X in the exponent of e should be x.)

III. Do Questions 6.3.1, 6.3.4, 6.3.6, 6.3.13, 6.3.14, 6.3.15 as you come to them in your reading.

IV. (Added  assignment passed out in class.) Optional but recommended: Try the demo at
 http://www.ruf.rice.edu/~lane/stat_sim/binom_demo.html

ASSIGNMENT FOR THURSDAY, APRIL 23

I. Do Questions 6.3.17, 6.3.19

II. Read Section 6.4

III. Do Questions 6.4.3 and 6.4.5 as you come to them in your reading.

IV. Turn in: Questions 6.3.2, 6.3.8, 6.3.16 (<- Note correction to what was handed out in class), plus:

1. A window shade will fit your window if its width is between 41.5 and 42.5 inches. You buy a blind from a shade from a brand that has average width 42” with standard deviation 0.25. What can you say about the probability that it will fit your window? [Hint: Chebyshev’s  inequality.]

 NO ASSIGNMENT TUESDAY, APRIL 28

THURSDAY, APRIL 30: EXAM (USUAL CLASSROOM)

ASSIGNMENT FOR  TUESDAY, MAY 5

1. Read:  Section 7.1, Definition 7.4.1 (p. 366), Theorem 7.4.2 (p. 367), and from the middle of p. 368 to the middle of p. 371.

2. Do Questions 7.4.2 (You will need to calculate the variance of Y) and 7.4.3 (You need the second corollary to the Bienayme-Chebyshev inequality.)

3. Try the Central Limit Theorem Java Applet at the website http://www.stat.sc.edu/~west/javahtml/CLT.html. (There’s a link from the class homepage.) One hundred or more rolls are most interesting, but 10,000 rolls sometimes  take a while to run.

ASSIGNMENT FOR THURSDAY, MAY 7

Hand in: Questions 6.3.24 and 6.4.4, plus the following problems:

1. A physical quantity is measured 50 times and the average of these measurements is taken as the result. If each measurement has a random  error uniformly distributed over the interval (-1, 1), what is the probability that the result differs from the actual value by less than 0.25?

2. Suppose that whenever invited to a party, the probability that a person attends with his or her guest is 1/3, attends alone is 1/3, and doesn’t attend it 1/3. A company has invited all 300 of its employees  to attend and bring a guest to a holiday party. What  is the probability that at least 320 people will attend?