• Author / Uploaded
• Yozef Pazmiño

Citation preview

´ DEPARTAMENTO DE ELECTRICA Y ´ ELECTRONICA

Carrera de Ingenier´ıa en Electr´onica y Telecomunicaciones Carrera de Ingenier´ıa en Electr´onica, Automatizaci´on y Control

´ PROCESOS ESTOCASTICOS

DEBERES

Dr. Enrique V. Carrera

SANGOLQU´I, ECUADOR 2018

Contents 1 Set Theory 1.1 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1

2 Combinatorics 2.1 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 3

3 Basic Probability 3.1 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 5

4 Conditional Probability 4.1 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 7

5 Probability 5.1 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 9

6 Random Variables 6.1 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 11

7 Probability Distributions 7.1 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 13

8 Random Variables 8.1 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 15

9 Multiple Random Variables 9.1 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 17

10 Functions of Random Variables 10.1 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 19

11 Random Processes 11.1 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20 20 20

i

12 Markov Processes 12.1 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22 22 22

13 Poisson Processes 13.1 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24 24 24

14 Analysis of Random Processes 14.1 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26 26 26

15 Estimation Theory 15.1 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28 28 28

16 Decision Theory 16.1 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30 30 30

17 Queueing Theory 17.1 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 31

ii

Homework 1

Set Theory∗ 1.1

Instructions

Solve the exercises listed in section 1.2 and write your report using LATEX† . After you generate the PDF file, you must submit your report through the on-line learning platform.

1.2

Exercises

1. Let A = {1, 0}. State whether or not each statement is correct: (i) {0} ∈ A, (ii) ∅ ∈ A, (iii) {0} ⊂ A, (iv) 0 ∈ A, (v) 0 ⊂ A. 2. Let U = {a, b, c, d, e, f, g}, A = {a, b, c, d, e}, B = {a, c, e, g} and C = {b, e, f, g}. Find: (i) A ∪ C; (ii) B ∩ A; (iii) C \ B; (iv) B c ∪ C; (v) C c ∩ A; (vi) (A \ C)c ; (vii) (A \ B c )c ; (viii) (A ∩ Ac )c . 3. In the Venn diagrams below, shade (i) W \ V , (ii) V c ∪ W , (iii) V ∩ W c , (iv) V c \ W c .

4. Let S = {a, b, c}, T = {b, c, d} and W = {a, d}. Construct the tree diagram of S × T × W and then find S × T × W . 5. Let A = B ∩ C. Determine if either statement is true: (i) A × A = (B × B) ∩ (C × C), (ii) A × A = (B × C) ∩ (C × B). 6. Let An = {x : x is a multiple of n} = {n, 2n, 3n, . . .}, where n ∈ N, the positive integers. Find: (i) A2 ∩ A7 ; (ii) A6 ∩ A8 ; (iii) A3 ∪ A12 ; (iv) A3 ∩ A12 ; (v) As ∪ Ast , where s, t ∈ N; (vi) As ∩ Ast , where s, t ∈ N. (vii) Prove: If J ⊂ N is infinite, then ∩i∈J Ai = ∅. 7. Find the power set P(A) of A = {1, 2, 3, 4} and the power set P(B) of B = {1, {2, 3}, 4}. 8. Find all partitions of V = {1, 2, 3}. ∗

Exercises based on the book Schaum’s Outline of Probability by Seymour Lipschutz and Marc L. Lipson (McGraw Hill, 2nd Ed., 2011). † There are many different TEX distributions, but I suggest to install TEXstudio (http://www.texstudio.org/) for Windows, Linux or MacOS. However, if you do not want to worry about software installation, I strongly recommend to use cloud applications like Overleaf (https://www.overleaf.com/).

1

9. Let [A1 , A2 , . . . , Am ] and [B1 , B2 , . . . , Bn ] be partitions of a set X. Show that the collection of sets [Ai ∩ Bj : i = 1, . . . , m, j = 1, . . . , n] is also a partition (called the cross partition) of X. 10. Let A and B be algebras (σ-algebras) of subsets of U . Prove that the intersection A ∩ B is also an algebra (σ-algebra) of subsets of U .

2

Homework 2

Combinatorics∗ 2.1

Instructions

Solve the exercises listed in section 2.2 and write your report using LATEX. After you generate the PDF file, you must submit your report through the on-line learning platform.

2.2

Exercises

1. (i) Find the number of ways in which five persons can sit in a row. (ii) How many ways are there if two of the persons insist on sitting next to one another? 2. How many different signals, each consisting of 8 flags hung in a vertical line, can be formed from 4 red flags, 2 blue flags and 2 green flags? 3. An urn contains 10 balls. Find the number of ordered samples (i) of size 3 with replacement,(ii) of size 3 without replacement, (iii) of size 4 with replacement, (iv) of size 5 without replacement. 4. (i) Find the number of distinct permutations that can be formed from all of the letters of the word ELEVEN. (ii) How many of them begin and end with E? (iii) How many of them have the 3 E’s together? (iv) How many begin with E and end with N? 5. A student is to answer 10 out of 13 questions on an exam. (i) How many choices has he? (ii) How many if he must answer the first two questions? (iii) How many if he must answer the first or second question but not both? (iv) How many if he must answer exactly 3 of the first 6 questions? (v) How many if he must answer at least 3 of the first 5 questions? 6. There are 12 balls in an urn. In how many ways can 3 balls be drawn from the urn, four times in succession, all without replacement? 7. In how many ways can 14 men be partitioned into 6 committees where 2 of the committees contain 3 men and the others 2? 8. Construct the tree diagram for the number of permutations of {a, b, c, d}. 9. A man has time to play roulette five times. He wins or loses a dollar at each play. The man begins with two dollars and will stop playing before the five times if he loses all his money or wins three dollars (i.e., has five dollars). Find the number of ways the playing can occur. ∗ Exercises based on the book Schaum’s Outline of Probability by Seymour Lipschutz and Marc L. Lipson (McGraw Hill, 2nd Ed., 2011).

3

10. Consider the following diagram with nine points A, B, C, R, S, T, X, Y, Z. A man begins at X and is allowed to move horizontally or vertically, one step at a time. He stops when he cannot continue to walk without reaching the same point more than once. Find the number of ways he can take his walk, if he first moves from X to R (by symmetry, the total number of ways is twice this).

4

Homework 3

Basic Probability∗ 3.1

Instructions

Solve the exercises listed in section 3.2 and write your report using LATEX. After you generate the PDF file, you must submit your report through the on-line learning platform.

3.2

Exercises

1. A die is weighted so that the even numbers have the same chance of appearing, the odd numbers have the same chance of appearing, and each even number is twice as likely to appear as any odd number. Find the probability that (i) an even number appears, (ii) a prime number appears, (iii) an odd number appears, (iv) an odd prime number appears. 2. Find the probability of an event if the odds that it will occur are (i) 2 to 1, (ii) 5 to 11. 3. Of 10 girls in a class, 3 have blue eyes. If two of the girls are chosen at random, what is the probability that (i) both have blue eyes, (ii) neither has blue eyes, (iii) at least one has blue eyes? 4. Ten students, A, B, . . ., are in a class. If a committee of 3 is chosen at random from the class, find the probability that (i) A belongs to the committee, (ii) B belongs to the committee, (iii) A and B belong to the committee, (iv) A or B belongs to the committee. 5. A class consists of 6 girls and 10 boys. If a committee of 3 is chosen at random from the class, find the probability that (i) 3 boys are selected, (ii) exactly 2 boys are selected, (iii) at least one boy is selected, (iv) exactly 2 girls are selected. 6. Three boys and 3 girls sit in a row. Find the probability that (i) the 3 girls sit together,(ii) the boys and girls sit in alternate seats. 7. A point is selected at random inside an equilateral triangle whose side length is 3. Find the probability that its distance to any corner is greater than 1. 8. A coin of diameter 21 is tossed randomly onto the Cartesian plane R2 . Find the probability that the coin does not intersect any line whose equation is of the form (i) x = k, (ii) x + y = k, (iii) x = k or y = k. Here k is an integer. 9. A point X is selected at random from a line segment AB with midpoint O. Find the probability that the line segments AX, XB and AO can form a triangle. ∗

Exercises based on the book Schaum’s Outline of Probability by Seymour Lipschutz and Marc L. Lipson (McGraw Hill, 2nd Ed., 2011).

5

10. Let A and B be events with P (A) = 12 , P (A ∪ B) = P (Ac ∩ B c ), P (Ac ∪ B c ) and P (B ∩ Ac ).

6

3 4

and P (B c ) = 85 . Find P (A ∩ B),

Homework 4

Conditional Probability∗ 4.1

Instructions

Solve the exercises listed in section 4.2 and write your report using LATEX. After you generate the PDF file, you must submit your report through the on-line learning platform.

4.2

Exercises

1. Four persons, called North, South, East and West, are each dealt 13 cards from an ordinary deck of 52 cards. (i) If South has exactly one ace, what is the probability that his partner North has the other three aces? (ii) If North and South together have 10 hearts, what is the probability that either East or West has the other 3 hearts? 2. In a certain town, 40% of the people have brown hair, 26% have brown eyes, and 16% have both brown hair and brown eyes. A person is selected at random from the town. (i) If he has brown hair, what is the probability that he also has brown eyes? (ii) If he has brown eyes, what is the probability that he does not have brown hair? (iii) What is the probability that he has neither brown hair nor brown eyes? 3. In a certain college, 25% of the boys and 10% of the girls are studying mathematics. The girls constitute 60% of the student body. If a student is selected at random and is studying mathematics, determine the probability that the student is a girl. 4. We are given two urns as follows: Urn A contains 5 red marbles, 3 white marbles and 8 blue marbles. Urn B contains 3 red marbles and 5 white marbles. A fair die is tossed; if 3 or 6 appears, a marble is chosen from B, otherwise a marble is chosen from A. Find the probability that (i) a red marble is chosen, (ii) a white marble is chosen, (iii) a blue marble is chosen. 5. Box A contains nine cards numbered 1 through 9, and box B contains five cards numbered 1 through 5. A box is chosen at random and a card drawn; if the card shows an even number, another card is drawn from the same box; if the card shows an odd number, a card is drawn from the other box. (i) What is the probability that both cards show even numbers? (ii) If both cards show even numbers, what is the probability that they come from box A? (iii) What is the probability that both cards show odd numbers? 6. A box contains three coins, two of them fair and one two-headed. A coin is selected at random and tossed. If heads appears the coin is tossed again; if tails appears, then another ∗

Exercises based on the book Schaum’s Outline of Probability by Seymour Lipschutz and Marc L. Lipson (McGraw Hill, 2nd Ed., 2011).

7

coin is selected from the two remaining coins and tossed. (i) Find the probability that heads appears twice. (ii) If the same coin is tossed twice, find the probability that it is the two-headed coin. (iii) Find the probability that tails appears twice. 7. A box contains 5 radio tubes of which 2 are defective. The tubes are tested one after the other until the 2 defective tubes are discovered. What is the probability that the process stopped on the (i) second test, (ii) third test? (iii) If the process stopped on the third test, what is the probability that the first tube is non-defective? 8. Urn A contains 5 red marbles and 3 white marbles, and urn B contains 2 red marbles and 6 white marbles. (i) If a marble is drawn from each urn, what is the probability that they are both of the game color? (ii) If two marbles are drawn from each urn, what is the probability that all four marbles are of the same color? 9. The probability that A hits a target is 41 and the probability that B hits a target is 13 . (i) If each fires twice, what is the probability that the target will be hit at least once? (ii) If each fires once and the target is hit only once, what is the probability that A hit the target? (iii) If A can fire only twice, how many times must B fire so that there is at least a 90% probability that the target will be hit? 10. A rifleman hits (H) his target with probability 0.4, and hence misses (M) with probability 0.6. He fires four times. (i) Determine the elements of the event A that the man hits the target exactly twice; and find P (A). (ii) Find the probability that the man hits the target at least once.

8

Homework 5

Probability∗ 5.1

Instructions

Solve the exercises listed in section 5.2 and write your report using LATEX. After you generate the PDF file, you must submit your report through the on-line learning platform.

5.2

Exercises

1. Show that (A ∩ B) \ C = (A \ C) ∩ (B \ C). 2. Show that F = {S, ∅} is an event space. 3. Let S = {1, 2, 3, 4} and F1 = {S, ∅, {1, 3}, {2, 4}}, F2 = {S, ∅, {1, 3}}. Show that F1 is an event space, and F2 is not an event space. 4. A random experiment has sample space S = {a, b, c}. Suppose that P ({a, c}) = 0.75 and P ({b, c}) = 0.6. Find the probabilities of the elementary events. 5. Let A, B, and C be three events in S. If P (A) = P (B) = 41 , P (C) = 13 , P (A ∩ B) = 81 , P (A ∩ C) = 16 , and P (B ∩ C) = 0, find P (A ∪ B ∪ C). 6. In an experiment consisting of 10 throws of a pair of fair dice, find the probability of the event that at least one double 6 occurs. 7. Show that if P (A) > P (B), then P (A|B) > P (B|A). 8. Show that (i) P (A∩B∩C) = P (A|B∩C)P (B|C)P (C), (ii) P (A∩B|C) = P (A|C)P (B|A∩ C). 9. There are 100 patients in a hospital with a certain disease. Of these, 10 are selected to undergo a drug treatment that 50 percent to 75 percent. What is the probability that the patient received a increases the percentage cured rate from drug treatment if the patient is known to be cured? 10. The relay network shown in next figure operates if and only if there is a closed path of relays from left to right. Assume that relays fail independently and that the probability of failure of each relay is as shown. What is the probability that the relay network operates? ∗

Exercises based on the book Schaum’s Outline of Theory and Problems of Probability, Random Variables, and Random Processes by Hwei P. Hsu (McGraw Hill, 2nd Ed., 2011).

9

10

Homework 6

Random Variables∗ 6.1

Instructions

Solve the exercises listed in section 6.2 and write your report using LATEX. After you generate the PDF file, you must submit your report through the on-line learning platform.

6.2

Exercises

1. Let X denote the number of heads obtained in the flipping of a fair coin twice. (i) Find the pmf of X. (b) Compute the mean and the variance of X. 2. Consider the function given by ( k/x2 pX (x) = 0

x = 1, 2, 3, . . . otherwise

where k is a constant. Find the value of k such that p(x) can be the pmf of a discrete r.v. X. 3. Given that X is a Poisson r.v. and pX (O) = 0.0498, compute E(X) and P (X ≥ 3). 4. A digital transmission system has an error probability of 10−6 per digit. Find the probability of three or more errors in 106 digits by using the Poisson distribution approximation. 5. The continuous r.v. X has the pdf ( k(2x − x2 ) fX (x) = 0

0 120); and (iii) P (10 < X ≤ 100). 7. Binary data are transmitted over a noisy communication channel in a block of 16 binary digits. The probability that a received digit is in error as a result of channel noise is 0.01. Assume that the errors occurring in various digit positions within a block are independent. (i) Find the mean and the variance of the number of errors per block. (ii) Find the probability that the number of errors per block is greater than or equal to 4. ∗ Exercises based on the book Schaum’s Outline of Theory and Problems of Probability, Random Variables, and Random Processes by Hwei P. Hsu (McGraw Hill, 2nd Ed., 2011).

11

8. In the manufacturing of computer memory chips, company A produces one defective chip for every nine good chips. Let X be time to failure (in months) of chips. It is known that 1 X is an exponential r.v. with parameter λ = 12 for a defective chip and λ = 10 with a good chip. Find the probability that a chip purchased randomly will fail before (i) six months of use; and (ii) one year of use. 9. Let the r.v. X denote the number of defective components in a random sample of n components, chosen without replacement from a total of N components, r of which are defective. The r.v. X is known as the hypergeometric r.v with parameters (N, r, n). (i) Find the pmf of X. (ii) Find the mean and variance of X. 10. Suppose the probability that a bit transmitted through a digital communication channel and received in error is 0.1. Assuming that the transmissions are independent events, find the probability that the third error occurs at the 10th bit.

12

Homework 7

Probability Distributions∗ 7.1

Instructions

Solve the exercises listed in section 7.2 and write your report using LATEX. After you generate the PDF file, you must submit your report through the on-line learning platform.

7.2

Exercises

1. Find (i) b(1; 5, 13 ), (ii) b(2; 7, 12 ), (iii) b(2; 4, 14 ). 2. The probability of a man hitting a target is 31 . (i) If he fires 5 times, what is the probability of hitting the target at least twice? (ii) How many times must he fire so that the probability of hitting the target at least once is more than 90%. 3. The mathematics department has 8 graduate assistants who are assigned to the same office. Each assistant is just as likely to study at home as in the office. How many desks must there be in the office so that each assistant has a desk at least 90% of the time? 4. Let X be a binomially distributed random variable with E(X) = 2 and V ar(X) = 43 . Find the distribution of X. 5. Let X be normally distributed with mean 8 and standard deviation 4. Find: (i) P (5 ≤ X ≤ 10), (ii) P (10 ≤ X ≤ 15), (iii) P (X ≥ 15), (iv) P (X ≤ 5). 6. Suppose the diameters of bolts manufactured by a company are normally distributed with mean 0.25 inches and standard deviation 0.02 inches. A bolt is considered defective if its diameter is ≤ 0.20 inches or ≥ 0.28 inches. Find the percentage of defective bolts manufactured by the company. 7. A fair die is tossed 720 times. Find the probability that the face 6 will occur (i) between 100 and 125 times inclusive, (ii) more than 150 times. 8. For the Poisson distribution p(k; λ), find (i) p(2; 1.5), (ii) p(3; 1), (iii) p(2; 0.6). 9. Suppose there is an average of 2 suicides per year per 50,000 population. In a city of 100,000 find the probability that in a given year there are (i) 0, (ii) 1, (iii) 2, (iv) 2 or more suicides. ∗

Exercises based on the book Schaum’s Outline of Probability by Seymour Lipschutz and Marc L. Lipson (McGraw Hill, 2nd Ed., 2011).

13

10. A die is ‘loaded’ so that the face 6 appears 0.3 of the time, the opposite face 1 appears 0.1 of the time, and each of the other faces appears 0.15 of the time. The die is tossed 6 times. Find the probability that (i) each face appears once, (ii) the faces 4, 5 and 6 each appears twice.

14

Homework 8

Random Variables∗ 8.1

Instructions

Solve the exercises listed in section 8.2 and write your report using LATEX. After you generate the PDF file, you must submit your report through the on-line learning platform.

8.2

Exercises

1. A fair coin is tossed four times. Let Y denote the longest string of heads occurring. Find the distribution, mean, variance and standard deviation of Y . 2. Two cards are selected at random from a box which contains five cards numbered 1, 1, 2, 2 and 3. Let X denote the sum and Y the maximum of the two numbers drawn. Find the distribution, mean, variance and standard deviation of (i) X, (ii) Y , (iii) X + Y , (iv) XY . 3. A box contains 10 transistors of which 2 are defective. A transistor is selected from the box and tested until a non-defective one is chosen. (i) Find the expected number of transistors to be chosen. (ii) Solve the problem in the case that 3 of the 10 items are defective. 4. The probability of team A winning any game is 21 . A plays team B in a tournament. The first team to win 2 games in a row or a total of three games wins the tournament. Find the expected number of games in the tournament. 5. A player tosses three fair coins. He wins $10 if 3 heads occur, $5 if 2 heads occur, $3 if 1 head occurs and $2 if no heads occur. If the game is to be fair, how much should he pay to play the game? 6. Consider the following joint distribution of X and Y . Find (i) E(X) and E(Y ), (ii) Cov(X, Y ), (iii) σX , σY and ρ(X, Y ). X \Y 1 5 Sum

−4 1/8 1/4 3/8

2 1/4 1/8 3/8

6 1/8 1/8 1/4

Sum 1/2 1/2 1

7. A fair coin is tossed four times. Let X denote the number of heads occurring and let Y denote the longest string of heads occurring. (i) Determine the joint distribution of X and Y . (ii) Find Cov(X, Y ) and ρ(X, Y ). ∗ Exercises based on the book Schaum’s Outline of Probability by Seymour Lipschutz and Marc L. Lipson (McGraw Hill, 2nd Ed., 2011).

15

8. Two cards are selected at random from a box which contains five cards numbered 1, 1, 2, 2 and 3. Let X denote the sum and Y the maximum of the two numbers drawn. (i) Determine the joint distribution of X and Y . (ii) Find Cov(X, Y ) and ρ(X, Y ). 9. Let X be a continuous random variable with distribution ( kx 0 ≤ x ≤ 5 fX (x) = 0 elsewhere (i) Evaluate k. (ii) Find P (1 ≤ X ≤ 3), P (2 ≤ X ≤ 4) and P (X ≤ 3). (iii) Determine and plot the graph of the cumulative distribution function FX (x). 10. If σX 6= 0, show that ρ(X, X) = 1 and ρ(X, −X) = −1.

16

Homework 9

Multiple Random Variables∗ 9.1

Instructions

Solve the exercises listed in section 9.2 and write your report using LATEX. After you generate the PDF file, you must submit your report through the on-line learning platform.

9.2

Exercises

1. Consider an experiment of tossing a fair coin three times. Let (X, Y ) be a bivariate r.v., where X denotes the number of heads on the first two tosses and Y denotes the number of heads on the third toss. (i) Find the range of X. (ii) Find the range of Y . (iii) Find the range of (X, Y ). (iv) Find P (X ≤ 2, Y ≤ 1), P (X ≤ 1, Y ≤ 1) and P (X ≤ 0, Y ≤ 0). 2. Let the joint pmf of (X, Y ) be given by ( k(xi + yj ) xi = 1, 2, 3; yj = 1, 2 pXY (xi , yj ) = 0 otherwise where k is a constant. (i) Find the value of k. (ii) Find the marginal pmf’s of X and Y . 3. The joint pdf of (X, Y ) is given by ( ke−(x+2y) fXY (x, y) = 0

x > 0, y > 0 otherwise

where k is a constant. (i) Find the value of k. (ii) Find P (X > 1, Y < 1), P (X < Y ), and P (X ≤ 2). 4. Let (X, Y ) be a bivariate r.v., where X is a uniform r.v. over (0, 0.2) and Y is an exponential r.v. with parameter 5, and X and Y are independent. (i) Find the joint pdf of (X, Y ). (ii) Find P (Y ≤ X). 5. The joint pdf of (X, Y ) is given by ( 2 2 xye−(x +y )/2 fXY (x, y) = 0

x > 0, y > 0 otherwise

(i) Find the marginal pdf’s of X and Y . (ii) Are X and Y independent? ∗

Exercises based on the book Schaum’s Outline of Theory and Problems of Probability, Random Variables, and Random Processes by Hwei P. Hsu (McGraw Hill, 2nd Ed., 2011).

17

6. The joint pdf of (X, Y ) is given by ( e−(x+y) fXY (x, y) = 0

x > 0, y > 0 otherwise

(i) Are X and Y independent? (ii) Find the conditional pdf’s of X and Y . 7. The joint pdf of (X, Y ) is given by ( e−y fXY (x, y) = 0

0 0, y > 0 fXY (x, y) = 0 otherwise (i) Find the joint moment generating function of X and Y . (ii) Find the joint moments m10 , m01 , and m11 . ∗ Exercises based on the book Schaum’s Outline of Theory and Problems of Probability, Random Variables, and Random Processes by Hwei P. Hsu (McGraw Hill, 2nd Ed., 2011).

19

Homework 11

Random Processes∗ 11.1

Instructions

Solve the exercises listed in section 11.2 and write your report using LATEX. After you generate the PDF file, you must submit your report through the on-line learning platform.

11.2

Exercises

1. Consider a random process X(t) defined by X(t) = Y cos(ωt + Θ) where Y and Θ are independent r.v.’s and are uniformly distributed over (−A, A) and (−π, π), respectively. (i) Find the mean of X(t). (ii) Find the autocorrelation function RX (t, s) of X(t). 2. Suppose that a random process X(t) is wide-sense stationary with auto-correlation RX (t, t+ τ ) = e−|τ |/2 . (i) Find the second moment of the r.v. X(5). (ii) Find the second moment of the r.v. X(5) − X(3). 3. Consider a random process X(t) defined by X(t) = U cos t + (V + 1) sin t, −∞ < t < ∞; where U and V are independent r.v.’s for which E(U ) = E(V ) = 0, E(U 2 ) = E(V 2 ) = 1. (i) Find the autocovariance function KX (t, s) of X(t). (ii) Is X(t) WSS? 4. Consider the random processes X(t) = A0 cos(ω0 t + Θ), Y (t) = A1 cos(ω1 t + Φ); where A0 , A1 , ω0 , and ω1 are constants, and r.v.’s Θ and Φ are independent and uniformly distributed over (−π, π). (i) Find the cross-correlation function of RXY (t, t + τ ) of X(t) and Y (t). (ii) Repeat (i) if Θ = Φ. 5. A certain product is made by two companies, A and B, that control the entire market. Currently, A and B have 60 percent and 40 percent, respectively, of the total market. Each year, A loses 32 of its market share to B, while B loses 12 of its share to A. Find the relative proportion of the market that each hold after 2 years. 6. Let X(t) be a Poisson process with rate λ. Find E{[X(t) − X(s)]2 } for t > s. 7. Let X(t) be a Poisson process with rate λ. Find P [X(t − d) = k|X(t) = j]

d > 0.

8. Let Tn denote the time of the nth event of a Poisson process with rate λ. Find the variance of Tn . ∗

Exercises based on the book Schaum’s Outline of Theory and Problems of Probability, Random Variables, and Random Processes by Hwei P. Hsu (McGraw Hill, 2nd Ed., 2011).

20

9. Consider a Markov chain with states {0, 1, 2} and  0 0.5 P = 0.5 0 1 0

transition probability matrix  0.5 0.5 0

Is state 0 periodic? 10. Assume that customers arrive at a bank in accordance with a Poisson process with rate λ = 6 per hour, and suppose that each customer is a man with probability 23 and a woman with probability 13 . Now suppose that 10 men arrived in the first 2 hours. How many woman would you expect to have arrived in the first 2 hours?

21

Homework 12

Markov Processes∗ 12.1

Instructions

Solve the exercises listed in section 12.2 and write your report using LATEX. After you generate the PDF file, you must submit your report through the on-line learning platform.

12.2

Exercises

1. Show that (cf + ce + de, af + bf + ae, ad + bd + bc) is a fixed point of the matrix   1−a−b a b  c 1−c−d d P = e f 1−e−f 2. A man’s smoking habits are as follows. If he smokes filter cigarettes one week, he switches to non-filter cigarettes the next week with probability 0.2. On the other hand, if he smokes non-filter cigarettes one week, there is a probability of 0.7 that he will smoke non-filter cigarettes the next week as well. In the long run, how often does he smoke filter cigarettes? 3. A gambler’s luck follows a pattern. If he wins a game, the probability of winning the next game is 0.6. However, if he loses a game, the probability of losing the next game is 0.7. There is an even chance that the gambler wins the first game. (i) What is the probability that he wins the second game? (ii) What is the probability that he wins the third game? (iii) In the long run, how often will he win? 4. For a Markov chain, the transition matrix is   0.5 0 0.5 0 0  P = 1 0.25 0.5 0.25 (2)

(2)

and the initial probability distribution is p(0) = ( 12 , 12 , 0). Find (i) p13 , (ii) p23 , (iii) p(2) , (2)

(iv) p1 , (v) the vector p(0) P n approaches, (vi) the matrix P n approaches. 5. Each year a man trades his car for a new car. If he has a Buick, he trades it for a Plymouth. If he has a Plymouth, he trades it for a Ford. However, if he has a Ford, he is just as likely to trade it for a new Ford as to trade it for a Buick or a Plymouth. In 1955 he bought his first car which was a Ford. (i) Find the probability that he has a a) 1957 Ford, b) 1957 Buick, c) 1958 Plymouth, d) 1958 Ford. (ii) In the long run, how often will he have a Ford? ∗ Exercises based on the book Schaum’s Outline of Probability by Seymour Lipschutz and Marc L. Lipson (McGraw Hill, 2nd Ed., 2011).

22

6. There are 2 white marbles in urn A and 4 red marbles in urn B. At each step of the process a marble is selected from each urn, and the two marbles selected are interchanged. Let X, be the number of red marbles in urn A after n interchanges. (i) Find the transition matrix P . (ii) What is the probability that there are 2 red marbles in urn A after 3 steps? (iii) In the long run, what is the probability that there are 2 red marbles in urn A? 7. A fair coin is tossed until 3 heads occur in a row. Let X, be the length of the sequence of heads ending at the nth trial. What is the probability that there are at least 8 tosses of the coin? 8. A player has 3 dollars. At each play of a game, he loses one dollar with probability 34 but wins two dollars with probability 14 . He stops playing if he has lost his 3 dollars or he has won at least 3 dollars. (i) Find the transition matrix of the Markov chain. (ii) What is the probability that there are at least 4 plays to the game? 9. The diagram on the right shows four compartments with doors leading from one to another. A mouse in any compartment is equally likely to pass through each of the doors of the compartment. Find the transition matrix of the Markov chain.

10. Draw a transition diagram for each transition matrix:     0 0.5 0.5 0.5 0.5 (ii) P = 0.25 0.25 0.5 (i) P = 0.3 0.7 0 0.5 0.5

23

Homework 13

Poisson Processes 13.1

Instructions

Solve the exercises listed in section 13.2 and write your report using LATEX. After you generate the PDF file, you must submit your report through the on-line learning platform.

13.2

Exercises

1. Let K(t) denote the number of customers arriving in the time interval (0, t). Time t is measured in minutes. We assume that the arrival process is a Poisson process with an arrival rate of λ = 50 customers per minute. Find the probabilities for the following events: (i) 6 customers arrive in time interval (0, 0.5), (ii) 15 customers arrive in time interval (0, 0.32), (iii) 10 customers arrive in time interval (0, 1.3), (iv) No one customer arrives in time interval (0, 0.002) 2. E-mails arrive at a server according to a Poisson process of rate 120 e-mails per minute. Find the probability that in a 1-minute period, 18 e-mails arrive during the first 10 seconds and 9 e-mails arrive during the last 5 seconds. 3. Patients arrive in a surgery according to a homogeneous Poisson process with intensity 6 patients an hour. The doctor starts to examine the patients only when the third patient arrives. (i) Compute the expected time from the opening of the surgery until the first patient starts to be examined. (ii) Compute the probability that in the first opening hour, the doctor does not start examining at all. 4. A hen wants to cross a one-way road, where cars drive according to a homogeneous Poisson process with intensity λ cars a time unit, all with the same speed. It takes c time units for the hen to cross the road. Assume that the hen starts to cross the road immediately when there is a chance to do it without being run over by a car. Compute the expected total waiting and crossing time for the hen. 5. In a casino, a bell rings every now and then. Each time the bell rings, the player can press a button. The player wins the game if they press the button before time 1 and, in addition, after they have pressed the button, the bell no longer rings until time 1. Assume that the bell rings according to a Poisson process with intensity λ. The strategy of the player is first to wait until time s and then to press the button immediately after the bell rings. (i) Find the probability that the player wins the game (depending on s)? (ii) Find the optimal value of s and the corresponding winning probability.

24

6. Passengers arrive at a railway station according to a homogeneous Poisson process with intensity λ. At the beginning (time 0), there are no passengers at the station. The train departs at time t. Denote by W be the total waiting time of all passengers arrived up to the departure of the train. Compute E(W ). 7. The mean number of bacteria per milliliter of a liquid is known to be 4. Assuming that the number of bacteria follows a Poisson distribution, find the probability that (i) in 1ml of liquid, there will be a) no bacteria b) 4 bacteria c) fewer than 3 bacteria, (ii) in 3ml of liquid there will be less than 2 bacteria, (iii) in 0.5ml of liquid there will be more than 2 bacteria. 8. In a large town, one person in 80, on average, has blood type X. If 200 blood donors are taken at random, find an approximation to the probability that they include at least five persons having blood type X. How many donors must be taken at random in order that the probability of including at least one donor of type X shall be 0.9 or more? 9. The number of calls coming per minute into a hotels reservation center is Poisson random variable with mean 3. (i) Find the probability that no calls come in a given 1 minute period. (ii)Assume that the number of calls arriving in two different minutes are independent. Find the probability that at least two calls will arrive in a given two minute period. 10. A new lightbulb is packaged in a wrapper with a statement that the ‘Life’ is 1000 hours. Interpreting the lifetime as an exponential chance variable, (i) what is the probability that the lifetime of your lightbulb is at least 1000 hours? (ii) What is the probability that the lifetime is less than 500 hours? (iii) Suppose that two such new lightbulbs are placed into a lamp. What is the probability that both of them have lifetimes less than 500 hours?

25

Homework 14

Analysis of Random Processes∗ 14.1

Instructions

Solve the exercises listed in section 14.2 and write your report using LATEX. After you generate the PDF file, you must submit your report through the on-line learning platform.

14.2

Exercises

1. Let X(t) be a WSS random process, and let X 0 (t) be its m.s. derivative. Show that E[X(t)X 0 (t)] = 0. 2. Let X(t) = A cos(ω0 t+Θ), where A and ω0 are constants, Θ is a uniform r.v. over (−π, π). Find the power spectral density of X(t). 3. A random process Y (t) is defined by Y (t) = AX(t) cos(ωc t + Θ), where A and ωc are constants, Θ is a uniform r.v. over (−π, π), and X(t) is a zero-mean WSS random process with the autocorrelation function RX (τ ) and the power spectral density SX (ω). Furthermore, X(t) and Θ are independent. Show that Y (t) is WSS, and find the power spectral density of Y (t). 4. Let X(t) and Y (t) be defined by X(t) = U cos ω0 t + V sin ω0 t, Y (t) = V cos ω0 t − U sin ω0 t, where ω0 is constant and U and V are independent r.v.’s both having zero mean and variance σ 2 . (i) Find the cross-correlation function of X(t) and Y (t). (ii) Find the cross power spectral density of X(t) and Y (t). 5. Let Y (t) = X(t) + W (t), where X(t) and W (t) are orthogonal and W (t) is a white noise. Find the autocorrelation function of Y (t). 6. A zero-mean WSS random process X(t) is called band-limited white noise if its spectral density is given by ( N0 /2, |ω| ≤ ωB SX (ω) = 0, |ω| > ωB Find the autocorrelation function of X(t). 7. A WSS random process X(t) is applied to the input of an LTI system with impulse response h(t) = 3e−2t µ(t). Find the mean value of Y (t) of the system if E[X(t)] = 2. ∗

Exercises based on the book Schaum’s Outline of Theory and Problems of Probability, Random Variables, and Random Processes by Hwei P. Hsu (McGraw Hill, 2nd Ed., 2011).

26

8. The input X(t) to the RC filter shown below is a white noise with average power σ 2 . Find the mean-square value of Y (t).

9. Suppose that the input to the filter shown below is a white noise with average power σ 2 . Find the power spectral density of Y (t).

10. Suppose that the input to the discrete-time filter shown below is a discrete-time white noise with average power σ 2 . Find the power spectral density of Y (n).

27

Homework 15

Estimation Theory∗ 15.1

Instructions

Solve the exercises listed in section 15.2 and write your report using LATEX. After you generate the PDF file, you must submit your report through the on-line learning platform.

15.2

Exercises

1. Let (X1 , . . . , Xn ) be a random sample of a Bemoulli r.v. X with pmf f (x; p) = px (1 − p)1−x , x = 0, 1, where p, 0 ≤ p ≤ 1, is unknown. Find the maximum-likelihood estimator of p. 2. The values of a random sample, 2.9, 0.5, 1.7, 4.3, and 3.2, are obtained from a r.v. X that is uniformly distributed over the unknown interval (a, b). Find the maximum-likelihood estimates of a and b. 3. In analyzing the flow of traffic through a drive-in bank, the times (in minutes) between arrivals of 10 customers are recorded as 3.2, 2.1, 5.3, 4.2, 1.2, 2.8, 6.4, 1.5, 1.9, and 3.0. Assuming that the inter-arrival time is an exponential r.v. with parameter λ, find the maximum likelihood estimate of λ. 4. Let (X1 , . . . , Xn ) be the random sample of a normal r.v. X with known mean µ and unknown variance σ 2 . Find the maximum likelihood estimator of σ 2 . 5. Let (X1 , . . . , Xn ) be the random sample of a normal r.v. X with mean µ and variance σ 2 , where µ is unknown. Assume that µ is itself to be a normal r.v. with mean µ1 and variance σ12 . Find the Bayes’ estimate of µ. 6. Let (X1 , . . . , Xn ) be the random sample of a normal r.v. X with variance 100 and unknown µ. What sample size n is required such that the width of 95 percent confidence interval is 5? 7. Continue to generate standard normal random variables until you have generated n of √ them, where n ≥ 100 is such that S/ n < 0.1, where S is the sample standard deviation of the n data values. (i) How many normals do you think will be generated? (ii) How many normals did you generate? (iii) What is the sample mean of all the normals generated? (iv) What is the sample variance? (v) Comment on the results of iii and iv. Were they surprising? ∗

Exercises based on the books Schaum’s Outline of Theory and Problems of Probability, Random Variables, and Random Processes by Hwei P. Hsu (McGraw Hill, 2nd Ed., 2011) and Introduction to Probability Models by Sheldon M. Ross (Elsevier, 10th Ed., 2010).

28

R1 2 8. Estimate 0 ex dx by generating random numbers. Generate at least 100 values and stop R1 when the standard deviation of your estimator is less than 0.01. Note that 0 g(x)dx = E[g(U )]. 9. Use the approach of approximating by integration to obtain an interval of size less than 0.1, which we can assert, with 95 percent confidence, contains π. How many runs were necessary? 10. To estimate θ, we generated 20 independent values having mean θ. If the successive values obtained were: 102, 93, 112, 111, 131, 124, 107, 122, 114, 136, 95, 141, 133, 119, 145, 122, 139, 151, 117, 143. How many additional random variables do you think we will have to generate if we want to be 99 percent certain that our final estimate of θ is correct to within ±0.5?

29

Homework 16

Decision Theory∗ 16.1

Instructions

Solve the exercises listed in section 16.2 and write your report using LATEX. After you generate the PDF file, you must submit your report through the on-line learning platform.

16.2

Exercises

1. Let (X1 , . . . , Xn ) be a random sample of a Bemoulli r.v. X with pmf f (x; p) = px (1 − p)1−x , x = 0, 1, where it is known that 0 < p ≤ 21 . Let H0 : pP= 12 , H1 : p = p1 (< 12 ), and n = 20. As a decision test, we use the rule to reject H0 if ni=1 xi ≤ 6. (i) Find the power function g(p) of the test. (ii) Find the probability of a Type I error α. (iii) Find 1 the probability of a Type II error β when a) p1 = 41 , b) p1 = 10 . 2. Let (X1 , . . . , Xn ) be a random sample of a normal r.v. X with mean µ and variance 36. Let H0 : µ = 50, H1 : µ = 55. As a decision test, we use the rule to accept H0 if x ¯ < 53, where x ¯ is the value of the sample mean. (i) Find the expression for the critical region R1 . (ii) Find α and β for n = 16. 3. Let (X1 , . . . , Xn ) be a random sample of a normal r.v. X with mean µ and variance 100. Let H0 : µ = 50, H1 : µ = 55. As a decision test, we use the rule that we reject H0 if x ¯ ≥ c. Find the value of c and sample size n such that α = 0.025 and β = 0.05. 4. Let X be a normal r.v. with zero mean and variance σ 2 . Let H0 : σ 2 = 1, H1 : σ 2 = 4. Determine the maximum likelihood test. 5. Let X be a normal r.v. with zero mean and variance σ 2 . Let H0 : σ 2 = 1, H1 : σ 2 = 4. Let P (H0 ) = 32 and P (H1 ) = 13 . Determine the MAP test. 6. Consider a binary decision problem with the following conditional pdf’s: f (x|H0 ) = 12 e−|x| and f (x|H1 ) = e−2|x| . Determine the Bayes’ test if P (H0 ) = 0.25, and the Bayes’ costs are C00 = C11 = 0, C01 = 1 and C10 = 2.

∗ Exercises based on the book Schaum’s Outline of Theory and Problems of Probability, Random Variables, and Random Processes by Hwei P. Hsu (McGraw Hill, 2nd Ed., 2011).

30

Homework 17

Queueing Theory∗ 17.1

Instructions

Solve the exercises listed in section 17.2 and write your report using LATEX. After you generate the PDF file, you must submit your report through the on-line learning platform.

17.2

Exercises

1. Customers arrive at the express checkout lane in a supermarket in a Poisson process with a rate of 15 per hour. The time to check out a customer is an exponential r.v. with mean of 2 minutes. (i) Find the average number of customers present. (ii) What is the expected idle delay time experienced by a customer? (iii) What is the expected time for a customer to clear a system? 2. Consider an M/M/1 queueing system. Find the probability of finding at least k customers in the system. 3. In a university computer center, 80 jobs an hour are submitted on the average. Assuming that the computer service is modeled as an M/M/1 queueing system, what should the service rate be if the average turnaround time (time at submission to time of getting job back) is to be less than 10 minutes? 4. The capacity of a communication line is 2000 bits per second. The line is used to transmit 8-bit characters, and the total volume of expected calls for transmission from many devices to be sent on the line is 12,000 characters per minute. Find (i) the traffic intensity, (ii) the average number of characters waiting to be transmitted, and (iii) the average transmission (including queueing delay) time per character. 5. A bank counter is currently served by two tellers. Customers entering the bank join a single queue and go to the next available teller when they reach the head of the line. On the average, the service time for a customer is 3 minutes, and 15 customers enter the bank per hour. Assuming that the arrivals process is Poisson and the service time is an exponential r.v., find the probability that a customer entering the bank will have to wait for service. 6. A post office has three clerks serving at the counter. Customers arrive on the average at the rate of 30 per hour, and arriving customers are asked to form a single queue. The average service time for each customer is 3 minutes. Assuming that the arrivals process ∗

Exercises based on the book Schaum’s Outline of Theory and Problems of Probability, Random Variables, and Random Processes by Hwei P. Hsu (McGraw Hill, 2nd Ed., 2011).

31

is Poisson and the service time is an exponential r.v., find (i) the probability that all the clerks will be busy, (ii) the average number of customers in the queue, and (iii) the average length of time customers have to spend in the post office. 7. Find the average number of customers L in the M/M/1/K queueing system when λ = µ. 8. A gas station has one diesel fuel pump for trucks only and has room for three trucks (including one at the pump). On the average, trucks arrive at the rate of 4 per hour, and each truck takes 10 minutes to service. Assume that the arrivals process is Poisson and the service time is an exponential r.v. (i) What is the average time for a truck from entering to leaving the station? (ii) What is the average time for a truck to wait for service? (iii) What percentage of the truck traffic is being turned away? 9. An air freight terminal has four loading docks on the main concourse. Any aircraft which arrive when all docks are full are diverted to docks on the back concourse. The average aircraft arrival rate is 3 aircraft per hour. The average service time per aircraft is 2 hours on the main concourse and 3 hours on the back concourse. How many additional docks are needed so that at least 80 percent of the arriving aircraft can be served in the main concourse with the addition of holding area? 10. A drive-in banking service is modeled as an M/M/1 queueing system with customer arrival rate of 2 per minute. It is desired to have fewer than 5 customers line up 99 percent of the time. How fast should the service rate be?

32