ECEn 370 Introduction to Probability

Transcription

1 RED- You can write on this exam. ECEn 370 Introduction to Probability Section 00 Final Winter, 2009 Instructor Professor Brian Mazzeo Closed Book Non-graphing Calculator Allowed No Time Limit IMPORTANT! WRITE YOUR NAME on every page of the exam. Answer questions -25 (Part I) on the provided bubble sheet. Questions -25 are worth point each. Answer questions (Part II) directly on the exam. Questions have their points specied individually. Do not discuss the exam with other students.

2 Part I. In the experiment consisting of a single roll of a strange die, the sample space consists of {, 3, 5, 6, 7, 0}. The probability law for the experiment is: P({}) = 8 P({3}) = 4 P({5}) = 4 P({6}) = 8 P({7}) = 8 P({0}) = 8 Evaluate the following: P({, 0}) + P({the roll is even})+p({the roll is negative}) A) /8 B) /4 C) 3/8 D) /2 E) 5/8 F) 3/4 G) 7/8 H) I) 9/8 2. A test for the probability-phobia-disease is assumed to be correct 99% of the time. We also assume that the probability of any random person in our class having this disease is Given that you just tested positive for this disease, what is the probability that you actually have the disease? A) B) C) D) E) F) None of the Above 3. Consider two independent fair coin tosses, and the following events: H = H 2 = D = {st toss is a head} {2nd toss is a head} {the two tosses have different results} Which of the following statements are true? ) H and H 2 are pairwise independent events. 2) H and D are pairwise independent events. 3) H 2 and D are pairwise independent events. 4) H and H 2 and D are independent events. A) only. B) and 2 only. C) and 3 only. D), 2, and 3 only. E) 2 and 3 only. F), 2, 3, and 4. G) 4 only. H) None of the Above. 2

3 4. A joint probability density function is given by the following: c, if 0 x and 0 y, f X,Y (x, y) = 5, if x 3 and y 3, 0, otherwise. What is the value of c that will make this a legitimate PDF? A) /5 B) 2/5 C) 3/5 D) 4/5 E) F) 0 G) None of the Above 5. Suppose the probability of at least one ash ood in Provo is 0.0 this year, the probability of at least one major blizzard in Provo is 0. this year, and the probability of at least one tornado in Provo is 0.05 this year. Assume that these disasters are independent. What is the probability that this year we will have at least one ood, blizzard, or tornado in Provo this year? A) 0.0 B) 0.02 C) 0.06 D) 0.5 E) 0.6 F) 0.7 G) 0.23 H) None of the Above 6. The time until a lightbulb burns out is modeled by an exponential variable X with an expected lifetime of year. What is the probability that the bulb will burn out between 2 and 3 years? A) B) C) D) E) F) None of the Above 7. On any given day a student's score on an exam is a normal random variable with an average of 70 and a standard deviation of 0. The student takes an exam today. What is the probability that the student scores between 60 and 90? Φ(y) is the cumulative distribution function of a standard normal random variable. A) Φ(30) B) Φ(3) C) Φ(90) Φ(60) D) Φ(2) ( Φ()) E) Φ(2) Φ() F) ( Φ(2)) Φ() G) Φ(90) ( Φ(60)) H) None of the Above 8. A random variable X has a probability density function given by { 2(x 2) f X (x) = 9, if 2 x 5, 0, otherwise. 3

4 Find the cumulative distribution function, F X (x), and compute the value of the following expression: A) 0 B) C) 3/9 D) 5/3 E) 7/9 F) 2 G) 3 H) None of the Above F X (0) + F X (4) + F X (6) 9. If at rst you don't succeed, try, try, try again. A computer will successfully send a message across a network with probability 0.7. The computer will retry sending the message until it is successfully sent. Given that we know that the computer will successfully transmit the message on or before the fourth attempt, what is the probability that the computer successfully sends the message on the rst attempt? A) B) C) D) E) F) None of the Above 0. The PDF of a random variable X is given by Compute E[X] + var(x). A) /2 B) C) 3/2 D) 2 E) 5/2 F) 3 G) 7/2 H) 9/2 I) None of the Above f X (x) = { 2 9x, 0 x 3, 0, otherwise.. Consider n independent tosses of a biased coin whose probability of heads Y, is uniformly distributed over the interval [0, ]. With X being the number of heads obtained, we see that E[X Y ] = ny and var(x Y ) = ny ( Y ). What is the variance of X? A) n 6 B) n C) n2 2 D) n + n 2 E) n+n2 6 F) n 6 + n2 2 G) n 6 + n2 2 + n3 8 H) n 2 + n2 6 + n3 2 I) None of the Above 4

5 2. Let the random variables X and Y be described by a joint PMF consisting of the points (0,0), (,), (,2), and (2,4) where each of these points has a probability of 4. Find the correlation coecient ρ(x, Y ). A) 3/34 B) 6/70 C) 32/33 D) 3/35 E) 34/35 F) 3/33 G) 3/35 H) 33/35 I) 32/35 3. Let X be a continuous random variable with a uniform PDF on the interval [0, ]. Let Y be a continuous random variable with a uniform PDF on the interval [0, 2]. Suppose you have a random variable Z = max{x, Y }. What is f Z ( 3 4 ) + f Z( 5 4 )? A) /4 B) /2 C) 3/4 D) E) 5/4 F) 3/2 G) 7/4 H) 2 I) None of the Above 4. Let X be a continuous random variable with a uniform PDF between 0 and 2. Let Y be a random variable described by Y = (X ) 2. Evaluate f Y ( 2 ) + f Y (2). A) /4 B) 2/4 C) /2 D) 2/2 E) 3/4 F) 3 2/4 G) H) + 2/4 I) 5/4 5

6 5. Random variables X, Y, and Z are independent from each other and have associated transforms M X (s) = λ λ s, M Y (s) = e λ(es ), and M Z (s) = 2 e2s + 2 e3s. If Q = X + Y Z + 2 then the associated transform for Q is: ) ( A) M Q (s) = λeλ(es λ s 2 e2s + 2 e3s) +) ( B) M Q (s) = λeλ(es λ+s 2 e2s + 2 e3s) ) ( C) M Q (s) = λeλ(es λ s 2 e 2s + 2 e 3s) s D) M Q (s) = λeλ (e ) ( λ+s 2 e2s + 2 e3s) s E) M Q (s) = λeλ (e ) ( 2 e 2s + 2 e 3s) λ+s ) F) M Q (s) = λe2λ(es λ s ) G) M Q (s) = λe2λ(es λ s H) None of the Above ( 2 e2s + 2 e3s) ( 2 e 2s + 2 e 3s) 6. Given that the transform of a particular random variable X is M X (s) = λ λ s, what is the third moment of the random variable X, E [ X 3]? A) λ B) λ 3 C) D) /λ E) 2/λ 2 F) 6/λ 2 G) 6/λ 3 H) 24/λ 4 I) 6/λ 4 7. The transform M X (s) is associated with a random variable X, and we assume that M X (s) is nite for all s in some interval [ a, a] where a is a positive number. The random variable Y is independent of X and has an associated transform M Y (s). Which of the following statements are true? ) M X (s) uniquely determines the CDF of X. 2) M X (s) uniquely determines the PDF of X. 3) M X (s) = esx f X (x)dx if X is continuous. 4) If Z = ax + b, then M Z (s) = e sb M X (as). 5) M X+Y (s) = M X (s)m Y (s). A) only. B) 3 only. C) 4 only. D), 3 and 4 only. E), 4, and 5 only. F) 2, 3, and 4 only. G), 2, 3, and 5 only. H), 3, 4, and 5 only. I) 2, 3, 4, and 5 only. J), 2, 3, 4, and You decide your student housing needs a makeover and decide to put in hardwood ooring (wow!). You go to your building products company and they tell you that their boards have a width of 3 inches with a tolerance of +/-.2 inch, which you assume gives you a uniform distribution for a single board from 2.8 to 3.2 inches. The width of your room is 20 inches. If you buy 4 boards and plan to lay them side by side, what is the probability that you will be able to cover the width of your room? Use Φ(z) to denote the CDF of the standard normal random variable. 6

7 A) Φ( ) B) Φ( 2) C) Φ( 3) D) Φ( 4) E) Φ( 5) F) Φ( 6) G) None of the Above 9. Consider the following statements. Which of them are true? Let Y, Y 2,... be a sequence of random variables.. If the sequence Y n converges with probability then it also converges in probability. 2. If the sequence Y n converges in probability then it also converges with probability. Suppose Y, Y 2,... are independent identically distributed random variables with common mean µ and variance σ Let S n = Y + Y Y n. The CDF of S n converges to the standard normal CDF. 4. Let M n = Y+Y2+ +Yn n. The CDF of M n converges to the standard normal CDF. 5. Let Z n = Y+Y2+ +Yn nµ σ. The CDF of Z n n converges to the standard normal CDF. A) only. B) 2 only. C) and 3 only. D) 2 and 3 only. E), 3, 4, and 5 only. F) 3, 4, and 5 only. G) and 4 only. H) and 5 only. I) 2 and 5 only. J), 2, and 5 only. 20. A source transmits a string of symobls consisting of 0 and s through a noisy communication channel. Symbols are received correctly except for the following errors: Given that a 0 is transmitted, a is received with probability 0.0. Given that a is transmitted, a 0 is received with probability Errors in dierent symbol transmissions are independent. If the string of symbols 00 is transmitted, what is the probability that it is received correctly? A) B) C) D) E) F) G) None of the Above 7

8 2. An internet service provider in a very small town has installed 0 modems to serve the needs of a population of 3 dialup customers. It is known that at a given time, each customer will need a modem connection with probability 0.6. What is the probability that there are more customers needing a connection than there are modems? A) B) 0.03 C) D) E) F) G) None of the Above. 22. You have a signal s = +. It is corrupted by the addition of noise, N, which is uniformly distributed over [ 2, 2]. Thus ( the ) received signal is s + N. What is the probability that the received signal is less than zero? 3 A) Φ 2 ( ) B) Φ 3 2 C) /8 D) /4 E) /2 F) 3/4 G) None of the Above 23. You work at a pizza shop where customers arrive as a Poisson process with a rate of λ = 0 customers per hour. Strangely, you work for two hours and no customers arrive. What is the probability that in the next two hours you get 40 customers? A) e B) e 20! C) 2e D) e E) 2e 20! F) e G) e H) ! ! ( e I) ( e ) ( e ! ) ( e ) ) 8

9 Questions 24 and 25 refer to the following Markov chain for the state of a student every year: 0.2 High School Senior Slacking Off 2 3 Studying Hard Unemployment Fast Food Job Engineering Job Promotion 24. Given that the student is in the Unemployment/Fast Food Job class, what is the probability that in any year the student is working at the Fast Food Job? A) /7 B) 3/4 C) 2/7 D) 5/4 E) 3/7 F) /2 G) 4/7 H) 9/4 I) 5/5 25. Given that a student is in state Slacking O, what is the probability that the student ends up in the class which contains an Engineering Job? A) 2/3 B) 6/3 C) 8/3 D) 0/3 E) 5/3 F) 7/3 G) 9/3 H) 2/3 I) 23/3 J) None of the above. 9

10 Part II 26. (4 points) Convergence Problem. a) In a small city, the quoted time for a paramedic to reach you after a 9 call is an average of 3 minutes. Without any other information, what is the upper bound on the probability that you call and it takes longer than 2 minutes for a paramedic to arrive? b) Suppose that you now know that the average arrival is 3 minutes and it has a standard deviation of /2 minute. What is the lower bound on the probability that after a call it arrives within 2 and 4 minutes? c) Suppose that the arrival time is normally distributed with an average time of 3 minutes and standard deviation of /2 minute. Suppose X and Y are random variables of the arrival times of two independent arrivals. Find the transforms of X and Y, M X (s) and M Y (s). Suppose that Z = X Y, the dierence of the two arrival times. Use transform methods to help you nd the PDF of Z. d) Suppose that you make 00 calls on separate occasions and the paramedic arrives 00 times with arrival times independent from each other, but with an average arrival time of 3 minutes and standard deviation of /2 minute. What is the probability that the total time the paramedic spends trying to get to you is more than 30 minutes? You can leave your answer in terms of Φ(z) where Φ is the standard normal CDF. 0

11 27. (4 points) The following Markov Chain represents an message transitioning between dierent servers in a computer network with unit time in each state. Source Server Lost Server Lost Server Server 2 Server Lost Server 6 7 Destination Server a) Given that a message starts at the source server, what is the probability that it will arrive at the destination server? b) What is the probability that you send ten messages and six of them reach the destination correctly? c) How many transitions would we expect it to take to go from the source to either the destination or lost classes? d) Suppose our message is in the lost server class. What is the probability that at a time far into the future (i.e. steady-state) that it will be found at lost server 6?