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Unformatted text preview: UNIVERSITY OF TEXAS AT AUSTIN EE 351K  P ROBABILITY & R ANDOM P ROCESSES I FALL 2010 EXAM 1 WEDNESDAY, SEPTEMBER 29, 2010 Name: Email: You have 75 minutes for this exam. The exam is closed book and closed notes. You are allowed to have one standard lettersized sheet, two sides, of handwritten notes. Calculators, laptop computers, Palm Pilots, twoway email pagers, etc. may not be used. Write your answers in the spaces provided. Please show all of your work. Answers without appropriate justification will receive very little credit. If you need extra space, use the back of the previous page. Problem 1 ( 16 pnts ): Problem 2 ( 12 pnts ): Problem 3 ( 12 pnts): Problem 1: (16 pnts) Alice has two boxes, blue and red, of chocolates; each box has n pieces of chocolate. Every day she eats one piece of chocolate. She picks this either from the blue box with probability p , or from the red box with probability 1 p ; the choices are independent from one day to the next. Let K denote the first day at which either one of the boxes first becomes empty i.e. the day its last chocolate is eaten. Note that this empty box could be either blue or red. K is a random variable. Find the PMF (probability mass function) of K . Hint: What is the probability of the event { K = n + k }{ Blue box first to be empty } Problem 2: (12 pnts) There are n balls and n bins. For each ball, a random bin is chosen and the ball is thrown into that bin all choices are independent. (a) (4 pnts) Let X be the number of bins with exactly one ball. Find E [ X ] . (b) (4 pnts) What is the probability that no bin is empty ? (c) (4 pnts) What is the probability that only one of the bins is occupied ? Problem 3: (12 pnts) A biological experiment starts with a single bacterium. At the end of one hour, it either splits into two with probability p , or dies with probability 1 p . If it splits, each of its children proceeds in the same way at the end of one more hour, each one either splits into two w.p. p or dies w.p. 1 p . And so on for their children, and their childrens children ... (a) (6 pnts) Let X be the number of bacteria at the end of two hours. Find the PMF of X . (b) (6 pnts) Suppose N is the random number of bacteria at some time, with PMF p N ( n ) . Let M be the random number one hour later. Find an expression for the PMF p M ( m ) in terms of p N ( n ) ....
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