Math 347/447 & ACSC 347      Probability & Statistics I      Spring 2009

Homework Problems & Assignments                        Revised 08/21/2009

Due dates are subject to change, as are the problems assigned more than one week in the future.  Supplemental problems may occasionally be assigned.
Homework problems: Work all assigned problems, which will be discussed at the beginning of class. Hand in the assigned even−numbered problems.

Assignments will be posted two weeks before they are due and links will appear below; they will not be handed out in class. This course carries both undergraduate (347) and graduate (447) credit. Each assignment will have one or two problems that will be required of graduate students and will be optional for undergraduates. Work must be shown in order to receive credit on the assignments. Assignments are to be handed in separately from the homework problems.
Please note that beginning with Chapter 3, all problems must be (re)stated using random variables, if possible.  The random variable must be defined at the beginning ("Let Y = the number of defective widgets" or "Let Y be the height of a randomly selected Martian," etc.) and its distribution must be stated, including any parameters ("Then Y is hypergeometric with parameters N=1000, r=20, n=5" or "Then Y is normal with mean 13 and variance 6" or ...). In problems like most of those in Sections 4.2 and 4.3, it will be enough to say something like "Let Y be a random variable with density function f(y) = ...."

Assignments

Link

Sections covered

Due:  6:00 p.m. on
Assignment 1 Chapter 2 February 23
Assignment 2 Sections 3.1−3.8 March 23
Assignment 3 Sections 3.8−4.2 March 30
Assignment 4 Sections 4.1−4.5 April 6
Assignment 5 Sections 4.6−6.3 May 4

Homework Problems (even numbers collected)
Date Due Section Page

Problems
February 2

#1

 2.3 25 2, 4, 5ac, 6.
2.4 32 9, 11, 14, 18, 21 (need Exercise 2.5a and c)
2.5 39 29, 32.
2.6 48 38, 41, 42.
February 9

#2 

 2.6 48 43, 44, 51, 53, 55, 57, 58, 59.
2.7 54 71, 74, 75, 76, 77.
2.8 59 86, 91, 93, 94, 95, 96, 101.
February 16

#3 

 2.9 68 114, 115, 116, 120ab.
2.10 72 124, 125, 130, 135, 136.
    Optional but instructive: applet problems 122 & 123; do not hand in. Click here to get the applet.
2 Suppl. 80 146, 147, 148, 149, 150, 162, 170.
February 23

#4

 3.2 90 2, 4, 7, 9.
3.3 97 Perform all calculations −− completely evaluate your answers: 12, 13, 14, 19.  Click here for a sample problem, the definition of variance, and the computing formula for the variance; use the computing formula for the variance in your calculations of the variance in these problems.
March 2 Test 1 Sections 2.1−3.3
March 2

#5

 3.3 99 30, 31, 34.
3.4 110 35,  38 [do not use Table 1 or the statistical functions
            on a calculator in problem 38], 
45 [You may use Table 1, pp. 839−841, for this and any later
      problems. Remember to begin: "Let Y be the number
      of cells activating the alarm when..."],      
52, 53.
3.5 119 69, 73, 74, 80.
March 9

#6

 3.4 113 60 [add part (e): Find the probability that at least 14 but not more than 18 survive].
3.6 &
3 Suppl.		 123
155		 90, 91.
209.
3.7 128 103, 105, 106, 110.
3.8 136 121, 122 (do not use tables or a calculator's pdf/cdf for #121a & 122c), 123, 124, 125, 130, 137.
March 23

#7

 3.9 142 147, 148, 149, 150, 153, 154 (explain how you get your answer in each part), 156, 157, 158, 159.
3.11 147 167, 170, 171.
3 Suppl. 153 195, 196, 197a, 199.
4.2 166 1, 7, 9, 11, 12 [part e should read "Find " ], 13, 18, 19.
March 30

#8

 4.3 172 22, 25, 26, 28, 32 [for part (c), you can actually calculate the probability that the cost exceeds $600.]
4.4 176

38, 41, 42, 44, 47, 51, 52. 
In 38b, calculate the probability to show that it depends only on the value of b (and not on the value of a).
4.5 181 58abcde (draw graphs), 59abc (draw graphs), 62, 68a, 69, 70 (add: what is the probability that exactly two of the three will have a gpa greater than 3.0?), 73.
April 6

#9

 4.5 184 74 [part (e) optional − don't hand in].
4.6 189 88, 89, 90, 97 & 102 (don't hand in), 98, 104, 106, 109, 110.
Hints:
     #90. This asks for a probability associated with a certain binomial with n = 10. Use the distribution in problem 4.69 to calculate the p of the binomial.
     #102. Click here to get the applet.
     #104. This asks for a probability associated with a certain binomial with n = 3. Use Y and f(y) to calculate the p of the binomial.
     #106 First find α and β from the given mean and variance.
     #109. To find V(L) = E(L2)–[E(L)]2, you need E(Y2),
E(Y3), E(Y4). The latter two can be found by writing the appropriate integral and "fudging" the constant to get the integral of a Gamma pdf, which will be 1.  Do not integrate by parts!  Or, look in the class notes, slide 7. But be able to do the "fudging."
     #110. Use the form of the density to determine the distribution and its parameters (and explain how you did this), and then use the table of distributions to get E(Y) and V(Y). Do not integrate.
4.7 197 123, 124, 125, 127.
    Optional but instructive: problems 115−119 (note that as of 2/12/09, there is a problem with the applet for problems 118 & 119 as noted on the site; hopefully this will be corrected); do not hand in. Click here to get the applet.
4.9 206 136, 137, 139 (hand in), 140, 141, 142, 145.
4.10 209 147.
April 13 Test 2 Sections 3.4−3.9, 4.2−4.6, 4.9
April 13

#10

 5.2 232 2, 5, 7, 16.
5.3 242 20 (also: find the conditional probability function for Y1
given Y2 = 1), 25, 34, 36.
April 20

#11 

 5.2 232 8 (click here for a hint for part b), 10 (both parts can be
done using geometry instead of integrating), 11 (use geometry), 13.
5.3 242 26, 38.
5.4 251 46, 51, 52, 54, 60, 67, 69.  No credit will be given for #46−60 without an explanation. All four can be explained in one sentence.

#2, #20, and #46 refer to the same (Y1,Y2);  
#7, #25, and #51 refer to the same (Y1,Y2);  
#8, #26, and #52 refer to the same (Y1,Y2);
#10 and #54 refer to the same (Y1,Y2);
#16, #36, and #60 refer to the same (Y1,Y2).
5.5−5.6 261 77 (refers to the distribution of #9 & #27, done in class), 78,
81 (refers to the distribution of #61, done in class), 82.
April 27

#12

 5.7 268 91, 95.
5.8 276 102, 103, 108, 111a, 112.
   #111a. Use the computing formula for the covariance to evaluate Cov(W1, W2), the numerator of .
   #112. You do not need to integrate. The joint support is a rectangle on which the joint density factors as g(y1) h(y2). From this you can determine the distributions of Y1 and Y2, using the tables in the back of the text. The factorization also tells you something else about Y1 and Y2.
6.3 307 1 (very good practice, especially if you have the Student Solution Manual), 6, 8, 10, 11, 13, 18.
May 4 Make−Up Test

Sections 3.1−3.9;  4.1−4.9.
You will be supplied with the distribution tables from the back cover of the book and both the text’s and the SOA’s normal tables.

You may use your calculator, but not the probability functions on the calculator.
  May 4

#13

 5.11 289 135, 137, 138, 139, 141.
   #135. Note that the number, Y, of defectives among the three has a binomial distribution with n = 3 and p ~ U(0, 1).
5 Suppl. 291 145, 153, 159.
6.4 316 23, 24, 30. 
When reading Section 6.4, I suggest you omit Examples 6.8 & 6.9.  These are more easily done using other methods.
6.5 322 39, 40 (see Example 6.11), 49, 52, 56, 59.
May 11 Final Exam

Sections 2.4−2.10;  3.1−3.9;  4.1−4.9;  5.1−5.8, 5.11;  6.3−6.5.

You will be supplied with the distribution tables from the back cover of the book and both the text’s and the SOA’s normal tables.

You may bring in a two−page crib sheet (either one two−sided sheet of paper or two sheets written on one side only).

You may use your calculator, but not the probability functions on the calculator.
  5.9 282 119, 122 (this is multinomial with n = 4 and k = 3 classes; the normal distribution of the mice weights is needed to calculate p1, p2, and p3), 126.

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