Homework 2. Spring 2019 (Due Thursday February 7)

Transcription

1 ECE 302: Probabilistic Methods in Electrical and Computer Engineering Spring 2019 Instructor: Prof. A. R. Reibman Homework 2 Spring 2019 (Due Thursday February 7) Homework is due on Thursday February 7 at the beginning of class. No late homework will be accepted. Include a brief description of all sources of information you used (including other people), not counting the text, handouts, or material posted on the web page, or state I did not receive help on this homework. You do not need to reference any material presented in class or on the course web-site, in the textbook, nor Prof. Reibman nor TA Chen Bai. Topics: Independent events; sequential experiments (Chapters 2.5, 2.6.1, 2.6.5). Random variables, PMF, PDF, and CDF; means and variances (Section , Sections , except Section 4.2.2) Exercise 1. A simplified model for the movement of the price of a stock supposes that on each day the stock s price either moves up one unit with probability p or it moves down one unit with probability 1 p. The changes on different days are assumed to be independent. (a) What is the probability that after two days the stock will be at its original price? (b) What is the probability that after three days the stock s price will have increased by one unit? (c) Given that after three days the stock s price has increased by one unit, what is the probability that it went up on the first day?

2 Exercise 2. (From Exam 1 Spring 2016; Note: this counts as 2 problems) Suppose 3 boxes contain Red, Green, and Blue marbles, denoted R, G, B, respectively. Box 1 has 3 Red, 4 Green, and 3 Blue. Box 2 has 8 Red, 1 Green, and 1 Blue. Box 3 has 0 Red, 4 Green, and 1 Blue. Suppose a box is chosen at random, and then a marble is selected from the box. (a) If box 1 is selected, what is the probability a Green marble is drawn? (b) What is the probability a Blue marble is drawn from Box 3? (c) What is the probability a Red marble is drawn? (d) Suppose a Red marble is drawn. What is the probability it came from Box 2? 2

3 Exercise 3. An experiment consists of picking one of two urns at random and then selecting a ball from the urn and noting its color (black or white). Let A be the event urn 1 is selected and B the event a black ball is observed. Under what conditions are A and B independent? (Hint: recall that independence is a mathematical definition, so determine what mathematical conditions are necessary, and how that translates into the world.) 3

4 Exercise 4. Let the sample space S = {ω 1, ω 2, ω 3 }, P (ω i ) = 1/3 for i = 1, 2, 3, and define random variables X, Y, and Z as: X(ω 1 ) = 1, X(ω 2 ) = 2 X(ω 3 ) = 3 Y (ω 1 ) = 2, Y (ω 2 ) = 3 Y (ω 3 ) = 1 Z(ω 1 ) = 3, Z(ω 2 ) = 1 Z(ω 3 ) = 2 (a) Show that these three random variables have the same probability mass function. (b) Find the pmf of X + Y, Y + Z, and Z + X. (c) Find the pmf of X + Y Z, Z/ X Y. 4

5 Exercise 5. (Textbook 3.20) Two dice are tossed and we let X be the absolute value of the difference in the number of dots facing up. (a) Find and sketch the pmf of X. (b) Find the probability that X k for all k. 5

6 Exercise 6. (From textbook, problems 4.13, 4.23, plus) A random variable X has cdf: { 0 for x < 0 F X (x) = e 2x for x 0 (a) Plot the cdf. Is this a discrete, continuous, or mixed RV? (b) Find and plot the pdf. (c) Use the cdf to find P (X 2), P (X = 0), P (X < 0), P (2 < X < 6). (d) Find mean and variance of X. 6

7 Exercise 7. (From textbook, problems 4.17, 4.39) A random variable X has pdf: { c(1 x f X (x) = 2 ) 1 x 1 0 otherwise (a) Find c and sketch the pdf. (b) Find and sketch the cdf of X. (c) Use the cdf to find P (X = 0), P (0 < X < 0.5), P ( X 0.5 < 0.25). (d) Find the mean and variance of X. 7

8 Exercise 8. A telephone installation has 2 lines, which allows zero, one, or two calls to happen simultaneously. The pmf of N, the number of active calls, is 0.2 n = n = 1 P N (n) = 0.1 n = 2 0 otherwise (a) Find E(N), the expected number of calls active. (b) Find E(N 2 ), the second moment of N. (c) Find V ar(n), the variance of N (d) Find σ N, the standard deviation of N. 8

9 Exercise 9. (Textbook 3.31) (a) Suppose a fair coin is flipped n times. Each coin toss costs d dollars and the reward in obtaining X heads is ax 2 + bx. Find the expected value of the net reward. (b) Suppose that the reward in obtaining X heads is instead a X for a > 0. Find the expected value of the reward. 9