INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
1st Markov Chain Homework
1. Suppose there are three white and three black balls in two urns distributed so that each urn contains three balls. We say the system is in state i, i = 0, 1, 2, 3, if there
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 3.4 Homework Answers
Homework: pgs. 105  106 #'s 1, 2, 8. 1. We consider one transmission. We let A be the event that a dot was transmitted. We let B be the event that a dot was received. Note
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 4.3 Homework Answers
Homework: pgs. 157158 #'s 1, 3, 4, 7, 8, 10. 1. F is given by 0 1/15 3/15 F (t) = 6/15 10/15 1 t<1 1t<2 2t<3 . 3t<4 4t<5 5t
3. We have R(X) = {2, 3, 4, 5,
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 4.2 Homework Answers
Homework: pgs. 150158 #'s 1, 4, 5, 6, 7, 16. 1. The possible values of X are {0, 1, 2, 3, 4, 5}. To find the probabilities of each, we note that all combinations of rolls
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 4.4 Homework Answers
Homework: pgs. 173  174, #'s 2, 3, 7, 11, 12. 2. Suppose the person chooses to park illegally. Let X be the amount of money he will pay on a given day. We have E[X] = 25
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 8.3 Homework Answers
Homework: pgs. 353  355, #'s 1, 3, 5 (only discrete case), 13. 1. We have that p(x, y) = We want, pXY (xy) = So we need to find pY (y) for y {0, 1, 2}.
2 1 (x2 25
+ y2
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
2nd Markov Chain Homework
1. Consider a Markov chain transition matrix 1/2 1/3 1/6 P = 3/4 0 1/4 . 0 1 0 (a) Show the this is a regular Markov chain. (b) If the process is started in state 1, find
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 9.1 Homework Answers
Homework: pgs. 383  384, #'s 1, 3, 5. 1. Let the number of hearts, diamonds, clubs, and spades be denoted by H, D, C, and S. We simply need to count the number of ways we
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 10.2 Homework Answers
Homework: pgs. 424  425, #'s 4, 5, 7, 9. 4. Let X and Y denote the numbers of sheep and goats stolen, respectively. We want Cov(X, Y ) = E[XY ]  E[X]E[Y ]. I will first
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 11.1 Homework Answers
Homework: pgs. 465  466, #'s 1, 3, 6, 7, 9, 17. 1. We have
5
MX (t) = E e
5
tX
=
x=1
etx P {X = x}
=
x=1
etx
1 5
=
1 t e + e2t + e3t + e4t + e5t . 5
3. We are g
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 10.3 Homework Answers
Homework: pgs. 433  434, #'s 1, 5, 6. 1. Cov(X, Y ) = (X, Y )X Y = (1/2) 2 3 = 3. Therefore, after first splitting off the "RV" 3, then splitting up the X and Y terms w
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 11.2 Homework Answers
Homework: pg. 474, #'s 2, 6, 8. 2. From number 9 of Section 11.1 (which you did), the moment generating function of a geometric RV, Xi , with parameter p is pet MXi (t) =
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 11.3 Homework Answers
Homework: pg. 484  485, #'s 1, 2, 3, 4, 5, 16 (k > 0). 1. Let X be the life of a given bill. We know that E[X] = 22, where time is in months. Thus, by Markov's inequality
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Section 11.4 HW Assignment Math331, Spring 2008 Instructor: David Anderson Hw: Consider a biased coin (say prob. of heads is p > 1/2) being flipped many times. Let Xi be one if a head appears on ith flip. Give a short writeup (roughly two paragraphs
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 11.5 Homework Answers
Homework: pg. 506, #'s 2, 8. 2. We have that X = (X1 + + X35 )/35. Let Z be a standard normal RV. Using the central limit theorem gives P (460 < X < 540) = P X1 +
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 3.5 Homework Answers
Homework: pg. 119 #'s 2, 4, 15, 17, 29 2. No, these events are not independent. Because they are old friends, we may assume that P (A) > 0 and P (B) > 0. However, P (AB) =
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 3.3 Homework Answers
Homework: pgs. 9697 #'s 1, 7, 9, 13. 1. Let A be the event that a given person is colorblind. Let M be the event that a person is male. Then, assuming that half the popula
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 3.2 Homework Answers
Homework: pgs. 87  88 #'s 1, 6, 10. 1. Let G be the event that Susan is guilty. Let L be the event that Robert lies. Then we are given that P (G) = .65, P (AG) = .25, and
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 4.6 Homework Answer
Homework: pgs. 185 # 2 2. Let X be the grade on the final that is comparable (after standardizing all of the RVs) to Velma's 82 on the midterm. Then we must have X  68 82 
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 5.1 Homework Answers
Homework: pgs. 196  197, #'s 1, 3, 4, 8, 13. 1. If we think of each draw as a Bernoulli RV with success defined as drawing a spade, the p = .25. Therefore, we see that if
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 5.2 Homework Answers
Homework: pgs. 211  212, #'s 3, 7, 11, 15. 3. n, the number of people in the theater, is 80. The probability that any one of them is an illegal immigrant is p = .025. Sinc
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 6.1 Homework Answers
Homework: pgs. 228  239, #'s 2, 3, 4. 2. (a) To find f , we simply need to differentiate F . This gives f (x) = 32x3 x 4 . 0 x<4
(c) The probability that the soap opera
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 8.1 Homework Answers
Homework: pgs. 325  326, #'s 1, 2, 3, 4, 5, 7. 1. We have that p(x, y) = (a) We must have
2 2 2
k
x y
0
if x = 1, 2, y = 1, 2 . else
1=
x=1 y=1
kx/y = k
x=1
x(1 + 1
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 5.3 Homework Answers
Homework: pgs. 224  226, #'s 3, 5, 9, 15, 17. 3. Let X be the number of times the professor tries the door until he is successful. Then we see that X is geometric with par
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 8.2 Homework Answers
Homework: pgs. 339  340, #'s 1, 2, 3, 5, 7, 8. 1. We have that 1 2 (x + y 2 ), x = 1, 2, y = 0, 1, 2, 25 and zero otherwise. Therefore, the marginal probability distributi
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
University of Wisconsin Math 331
February 15, 2008
Test 1
Name:
1. Legibly print your name above. 2. Do not open this test booklet until you are directed to do so. 3. Budget your time wisely! 4. This test is closed book and closed notes. You will
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
University of Wisconsin Math 331
February 15, 2008
Test 1
Name: Answer Key
1. Legibly print your name above. 2. Do not open this test booklet until you are directed to do so. 3. Budget your time wisely! 4. This test is closed book and closed notes
INTORUCTION TO PROBABILITY AND MARKOV CHAIN MODELS
MATH 331

Spring 2008
Math331, Spring 2008 Instructor: David Anderson
Section 1.1 and 1.2 Hw answers
Homework: pgs. 9  11, #'s 2, 3, 5, 9, 13, 17(a) 2. The sample space is given by S = {(i, j, k)  i, j, k {red,blue}. 3. The sample space is S = {t : t (0, 20)}. The ev