36-225 - Introduction to Probability Theory - Fall 2016 Homework 5 Due Friday, October 7th at Noon in the mailbox outside the BH 132 doorway Exercise numbers refer to Wackerly 7. If you are not using the American print edition of Wackerly 7: check the problem scans on Blackboard to ensure you are completing the correct exercise! Feel free to use R functions such as, e.g., dpois() to compute probabilities. Simply write down on your homework sheet the R function call, and be sure to include function call arguments. However, realize that you cannot use R on a test, so make sure you know how to compute any given probability by hand! (Thus it is best to use R to confirm your by-hand computations of probabilities, and not to replace such computations.) Test 1 will cover the hypergeometric and Poisson distributions, but not the uniform or normal distributions. Keep that in mind when reviewing these homework problems before the October 11th exam. 1) [10 pts] In your pocket is a random number of coins N , where N Poisson(). You toss each coin once, with heads showing with probability p each time. Show that the total number of heads has the Poisson distribution with parameter p.1 2) [7 pts] Exercise 3.112 3) [7 pts] Exercise 3.118 4) [7 pts] Exercise 3.124. Use the R function ppois() to estimate ; never use a table. 5) [7 pts] Exercise 3.128 6) [7 pts] Exercise 3.138 7) [10 pts] Exercise 3.140 8) [7 pts] Exercise 4.40 9) [7 pts] A r.v. Y is sampled from a piecewise constant distribution: 1/3 0 y < 1 2/3 1 y < 2 . f (y) = 0 otherwise (a) Compute E[Y ]. (b) Compute P (Y > 1|Y < 1.5). 10) [10 pts] A binary message is transmitted as a signal S, which we'll say can take on the values 1 or +1 with equal probability. The transmission channel corrupts the message with additive noise, i.e., the received value R = S + \u000f, where \u000f N (0, 2 ). The receiver assumes S = 1 if R < 0 and assumes S = +1 if R 0. What is the probability of the receiver making an error (i.e., what is the probability of S and R having opposite sign)? Assume S and \u000f are independent random variables. State your answer in terms of standard normal cdf function (). R 2 11) [7 pts] Show analytically that ex dx = , and thus show that \u0012 \u0013 1 (x )2 f (x) = exp 2 2 2 qR R R 2 is a density function if > 0. (Hint: note that ex dx = ex2 dx ey2 dy, which recasts the original integral into a double integral that you may find easier to work with . . . particularly if you do a coordinate transformation.)2 1 This 2 This is based on Problem 3.5.2 of Grimmett & Stirzaker, Probability and Random Processes, 2nd edition. is based on Problem 4.14.1 of Grimmett & Stirzaker, Probability and Random Processes, 2nd edition. NOTE: In problems 12-13, you will need to use R or some other calculating device to evaluate your answer. Do not use a probability table such as the one on the inside of the front cover of Wackerly 7! No one uses tables nowadays. 12) [7 pts] A manufacturer creates widgets whose mass has mean 5 and standard deviation . Ten widgets are shipped, and the shipment will be rejected if any of the widgets has mass 4 or 6. Give the largest value can be such that the probability of rejecting the shipment is 0.01. 13) [7 pts] Exercise 4.74 (but not part e) Be sure: To make this work your own even if you work with others! Do not blindly copy the solutions; this is plagarism, and furthermore it will only hurt you at exam time. To put your name on all sheets, and to staple the sheets together, if you are turning in multiple sheets. To properly label all plots (axes, main title, units, etc.) for full credit, if applicable, and to include a printout of any code used in solutions, if applicable