Math 347/447 & ACSC 347      Probability & Statistics I      Spring 2010

Homework Problems & Assignments                        Revised 03/20/2010

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 on Tuesdays. 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 extra credit 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.
Beginning with Chapter 3, all problems must be (re)stated using random variables, if possible.  The random variable must be defined at the beginning, such as:
     "Let Y = the number of defective widgets."
    "Let Y be the height of a randomly selected Martian."
You must also state its distribution, including all parameters, such as:
     "Then Y is hypergeometric with parameters N = 1000, r = 20, and n = 5."
    "Then Y is normal with mean 13 and variance 6."
In problems like most of those in Sections 4.2-4.3, it will be enough to say something like
    "Let Y be a random variable with density function f (y) = ...."

Assignments

Link (to Blackboard)

Sections covered

Due:  11:00 a.m. on
Assignment 1 Chapter 2 Thursday, February 25
Assignment 2 Sections 3.1-3.7 Thursday, March 25

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, 23, 34.  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.
    Optional, but instructive: 30, 31.
March 2

#5

 3.4 110 35,  38 [do not use Table 1 nor 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..."],      
51, 52, 53, 60 [add part (e): Find the probability that at
least 14 but not more than 18 survive].
3.5 119 69, 73, 74, 78 [Hint for 2nd part: see/use Exercise 3.71], 80.
March 4 Test 1 Sections 2.1−3.3
March 9

#6

 3.6 &
3 Suppl.		 123
155		 90, 91.
209.
3.7 128 103, 105, 106, 110.
Uniform Discrete Distribution Prove the theorem about the Expectation and Variance of a Uniform Discrete Distribution in the class lecture notes. This problem is to be handed in.
March 23

#7

 3.8 136

121, 122 (do not use tables or a calculator's pdf/cdf for #121a & 122c), 123, 125, 128, 129, 130, 137.

#130: To calculate the probability of the event A that exactly 3 cars arrive at the lot in a given hour,  partition A into 4 events according to how may arrive at each entrance. Calculate these probabilities using the given Poisson distributions and independence, and sum (why can you sum?).

#129: The answer in the text and the solution in the Student Solution manual are incorrect. The correct answer is ≈0.687 minutes or ≈41 seconds.
3.9 142

147, 148, 149, 150, 153, 154, 156, 158, 159.

#154: explain how you get your answer in each part. I found it convenient to do #153 and #154 together.

#156b and #158: start with the definition of the mgf of W,
mW(t) = E(etW), replace W by its expression in terms of Y, and then use the definition of the mgf of Y; you also need to use properties of expectations in #158. Similarly for #156c with X in place of W.
3 Suppl. 153

195, 196, 197a, 199.

#197a is an important type of problem, which we'll see again in Chapter 4. If Y is the number of colonies in a 1-cm3 sample, then Y has the given Poisson distribution. If 4 samples are selected and X is the number of them with at least 1 colony, then we want P(X ≥ 1). Then X has a certain binomial distribution and Y is used to calculate its parameter, p.
4.2 166

1, 9, 11, 12, 13.

#12a: See Theorem 4.1 on p. 160.
#12b: Solve F(y) = 0.30 for y = φ0.30.   See the definition and the example on slides 18-19 of the §4.2 notes.
#12e: It should read "Find ."

#13: This is similar to #4.16bd, which was done in class (slides 11-17, but there is a difference: there are two intervals on which f(y) is nonzero (and it is defined differently on the two). Example 3 in the class notes (stated on slide 22 and solved in the handout 4.02.Examples.pdf on the class website) is similar. Attempt #4.13 and make sure you correctly find F(y) -- the answer in the back of the text is correct. This type of problem comes up often (including in the next assignment).
  4.2 169 18, 19.
  4.3 172 22, 25, 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.

#38b: calculate the probability to show that it depends only on the value of b (and not on the value of a).
#42: use the formula for F(y) from slide 5 to find the median.
#44: do not use the formula for F(y) from slide 5 in part b.

  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.
  4.5 184 74 [part (e) is optional − don't hand it in].
  4.6 189

88, 89, 90, 97 & 102 (don't hand in), 98, 104, 106, 109, 110.

Hints:  Several (most?) of the problems which involve an exponential distribution are simpler if you use the formulas for
F(y) and S(y) from slide 11 of the Section 4.6 class notes.

     #90. This asks for a probability associated with a certain binomial with n = 10. Use the exponential distribution in problem 4.88, and its F(y), 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 to calculate the p of the binomial. What is the distribution of Y ?

     #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 115−117; do not hand in.
 Click here to get the applet.
  4.9 206 136, 137, 139 (hand in), 140, 141, 142, 145.
 

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