ECEn 370 Introduction to Probability

Transcription

1 ECEn 370 Introduction to Probability Section 001 Midterm Winter, 2014 Instructor Professor Brian Mazzeo Closed Book - You can bring one 8.5 X 11 sheet of handwritten notes on both sides. Graphing or Scientic Calculator Allowed 3-Hour Suggested Time Limit IMPORTANT! WRITE YOUR NAME on every page of the exam. Answer questions 1-25 on the provided bubble sheet. Questions 1-25 are worth 1 point each. Do not discuss the exam with other students. NOTE: Use all of the digits on your calculator, or fractions, before at the end rounding to the number of signicant digits used in the problem. 1

2 1. A conservative design team, call it C, and an innovative design team, call it N, are asked to separately design a new product within a month. From past experience we know that: a) The probability that team C is successful is 2/3. b) The probability that team N is successful is 1/2. c) The probability that at least one team is successful is 3/4. Assuming that exactly one successful design is produced, what is the probability that it was designed by team N? A) B) C) D) E) F) G) H) I) You enter a chess tournament where your probability of winning a game is 0.3 against half the players (call them type 1), 0.4 against a quarter of the players (call them type 2), and 0.5 against the remaining quarter of the players (call them type 3). You play a game against a randomly chosen opponent. What is the probability of winning? A) B) C) D) E) F) G) H) I) Suppose that you win the game from Problem 2. Given that information, what is the probability that you had an opponent of type 1? A) B) C) D) E) F) G) H) I)

3 4. A test for a certain rare disease is assumed to be correct 95% of the time: if a person has the disease, the test results are positive with probability 0.95, and if the person does not have the disease, the test results are negative with probability A random person drawn from a certain population has probability of having the disease. Given that the person just tested positive, what is the probability of having the disease? A) B) C) D) E) F) G) H) I) A computer network connects two nodes A and B through intermediate nodes C, D, E, F, as shown in the following gure. For every pair of directly connected nodes, say i and j, there is a given probability p ij that the link from i to j is up. We assume that link failures are independent of each other. What is the probability that there is a path connecting A and B in which all links are up? C E 0.9 A F B 0.75 D 0.95 A) B) C) D) E) F) G) H) I) An internet service provider has installed c modems to serve the needs of a population of n dialup customers. It is estimated that at a given time, each customer will need a connection with probability p, independent of the others. What is the probability that there are more customers needing a connection than there are modems? Suppose that n = 100, p = 0.91, and c = 97. A) B) C) D) E) F) G) H) I)

4 7. Let Y = X and let the PMF for X be given by { 1/9, if x is an integer in the range [-4, 4], p X (x) = 0, otherwise What is p Y (0) + p Y (1) + p Y (2)? A) 1/9 B) 2/9 C) 3/9 D) 4/9 E) 5/9 F) 6/9 G) 7/9 H) 8/9 I) 1 8. If the weather is good (which happens with probability 0.6), Alice walks the 2 miles to class at a speed of V = 5 miles per hour, and otherwise rides her motorcycle at a speed of V = 30 miles per hour. What is the mean of the time T to get to class (in hours)? A) 2/15 B) 3/15 C) 4/15 D) 5/15 E) 6/15 F) 7/15 G) 8/15 H) 9/15 I) 10/15 9. Consider a quiz game where a person is given two questions and must decide which one to answer rst. Question 1 will be answered correctly with probability 0.8, and the person will then receive as prize $100, while question 2 will be answered correctly with probability 0.5, and the person will then receive as prize $200. If the rst question attempted is answered incorrectly, the quiz terminates, i.e., the person is not allowed to attempt the second question. If the rst question is answered correctly, the person is allowed to attempt the second question. Assuming that you choose the best strategy (i.e., which question should be answered rst), what is the expected value of the total prize money received? A) $130 B) $140 C) $145 D) $150 E) $155 F) $160 G) $165 H) $170 I) $175 4

5 10. Consider the following gure of the joint PMF in tabular form for p X,Y (x, y). y 4 0 1/20 1/20 1/20 3 1/20 2/20 3/20 1/20 2 1/20 2/20 3/20 1/20 1 1/20 1/20 1/ x Find the marginal PMFs for X and Y and determine p Y (2) + p Y (4) + p X (2) + p X (3) + p X (5) A) 16/20 B) 17/20 C) 18/20 D) 19/20 E) 20/20 F) 21/20 G) 22/20 H) 23/20 I) 24/ Again consider the joint PMF from question 10. Suppose you have a random variable Z given by Z = X + 2Y. Find E[Z]. A) 7.55 B) 9.55 C) D) E) F) G) H) I) Let X be the roll of a fair six-sided die and let A be the event that the roll is an even number. What is the probability that we roll a 2 given that the roll is even? A) 1/12 B) 2/12 C) 3/12 D) 4/12 E) 5/12 F) 6/12 G) 7/12 H) 8/12 I) 9/12 5

6 13. Messages transmitted by a computer in Boston through a data network are destined for New York with probability 0.5, for Chicago with probability 0.3, and for San Francisco with probability 0.2. The transit time X of a message is random. Its mean is 0.05 seconds if it is destined for New York, 0.1 seconds if it is destined for Chicago, and 0.3 seconds if it is destined for San Francisco. What is E[X] in seconds? A) 0.05 B) 0.10 C) 0.15 D) 0.20 E) 0.25 F) 0.30 G) 0.35 H) 0.40 I) The time until a small meteorite rst lands anywhere in the Sahara desert is modeled as an exponential random variable with a mean of 10 days. The time is currently midnight. What is the probability that a meteorite rst lands some time between 6 a.m. and 6 p.m. of the rst day? A) B) C) D) E) F) G) H) I) You are allowed to take a certain test three times, and your nal score will be the maximum of the test scores. Thus, X = max{x 1, X 2, X 3 }, where X 1, X 2, X 3 are the three test scores and X is the nal score. Assume that your score in each test takes one of the values from 1 to 10 with equal probability 1/10, independent of the scores in other tests. What is p X (8)? A) B) C) D) E) F) G) H) I)

7 16. The annual snowfall at a particular geographic location is modeled as a normal random variable with a mean of µ = 60 inches and a standard deviation of σ = 20 inches. What is the probability that this year's snowfall will be at least 80 inches? Φ(y) is the CDF of the standard normal random variable. A) Φ( 1/2) B) Φ(1) C) 1 + Φ(1) D) Φ(1/2) E) Φ( 1/4) F) 1 Φ(1/2) G) Φ(2) H) Φ(3) I) 1 Φ(2) 17. Suppose I have a joint uniform PDF in the triangular area A described by the points (0, 0), (0, 1), (3,3). The PDF is described by: { c, in area A f X,Y (x, y) = 0, otherwise What is c? A) 1/6 B) 1/3 C) 1/2 D) 2/3 E) 5/6 F) 1 G) 7/6 H) 8/6 I) 3/2 18. For the PDF in problem 17, what is the CDFF X,Y (x, y) evaluated at point (1/2, 1/2)? A) 1/12 B) 2/12 C) 3/12 D) 4/12 E) 5/12 F) 6/12 G) 7/12 H) 8/12 I) 9/12 7

8 19. Let X and Y be described by a uniform PDF on the unit square (0 x, y 1). Which of the following describes the joint CDF? A) x + y B) x y C) xy D) xy 4 E) x/y F) y/x G) (x + 1)(y + 1) H) (x 1)(y 1) I) xy + 2x + 2y 20. The time T until a new light bulb burns out is an exponential random variable with parameter λ. Ariadne turns the light on, leaves the room, and when she returns, t time units later, nds that the light bulb is still on, which corresponds to the event A = {T > t}. Let X be the additional time until the light bulb burns out. What is the conditional CDF of X, given the event A? Evaluate this for the substitutions that x = 1, t = 2, and λ = 1. A) e 1 B) e 1 e 2 C) e 1 /e 2 D) 1 e 2 E) 1 e 2 /e 1 F) 1 e 1 G) e 2 /e 1/2 H) e 3 /e 1 I) e The metro train arrives at the station near your home every quarter hour starting at 6:00 a.m. You walk into the station every morning between 7:10 and 7:30 a.m., and your arrival time is a uniform random variable over this interval. If you nd the PDF of Y, which is your time to wait for the train to arrive, then evaluate f Y (3)+f Y (10)? A) 1/60 B) 2/60 C) 3/60 D) 4/60 E) 5/60 F) 7/60 G) 8/60 H) 9/60 I) 10/60 8

9 22. Ben throws a dart at a circular target of radius r = 2. We assume that he always hits the target, and that all points of impact (x, y) are equally likely, so that the joint PDF of the variables X and Y is uniform. Find the probability density f X,Y (x, y) and then evaluate the following: f X,Y (0, 0) + f X,Y (1, 1.5). A) B) C) D) E) F) G) H) I) The speed of a typical vehicle that drives past a police radar is modeled as an exponentially distributed random variable X with mean 50 miles per hours. The police radar's measurement Y of the vehicle's speed has an error which is modeled as a normal random variable with zero mean and standard deviation equal to one tenth of the vehicle's speed. Find the joint PDF, f X,Y (x, y) for X and Y. Then evaluate this joint PDF at the point (50, 51). A) B) C) D) E) F) G) H) I) A binary signal S is transmitted, and we are given that P (S = 1) = p and P (S = 1) = 1 p.the received signal is Y = N + S, where N is normal noise, with zero mean and unit variance, independent of S. What is the probability that S = 1, as a function of the observed value y of Y? Evaluate this for the case that p = 0.75 and y = 0.3 A) B) C) D) E) F) G) H) I)

10 25. The sample space of an experiment is given by S = {a, b, c, d} with probabilities P (a) = 0.2, P (b) = 0.3, P (c) = 0.4, and P (d) = 0.1. Let A denote the event {a, b}, and B the event {b, c, d}. Evaluate P (A B)+P (A B). A) 0.6 B) 0.7 C) 0.8 D) 0.9 E) 1.0 F) 1.1 G) 1.2 H) 1.3 I)