SomePracticeProblems

# SomePracticeProblems - UNIVERSITY OF TEXAS AT AUSTIN EE...

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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 letter-sized sheet, two sides, of handwritten notes. • Calculators, laptop computers, Palm Pilots, two-way e-mail 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 children’s 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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SomePracticeProblems - UNIVERSITY OF TEXAS AT AUSTIN EE...

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