hw3 - wins the match Each game is won by Fischer with...

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EE 351K Probability and Random Processes FALL 2010 Instructor: Haris Vikalo [email protected] Homework 3 Due on : Tuesday 09/21/10 Problem 1 There are three dice in a bag. One has one red face, another has two red faces, and the third has three red faces. One of the dice is drawn at random from the bag, each die having an equal chance of being drawn. The selected die is repeatedly rolled. What is the probability that red comes up on the first roll? Given that red comes up on the first roll, what is the conditional probability that red comes up on the second roll? Given that red comes up on the first three rolls, what is the conditional probability that the selected die has red on three faces? Problem 2 Let X be a discrete random variable that is uniformly distributed over the set of integers in the range [ a,b ] , where a and b are integers with a < 0 < b . Find the PMF of the random variables max { 0 ,X } and min { 0 ,X } . Problem 3 Fischer and Spassky play a sudden-death chess match whereby the first player to win a game
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Unformatted text preview: wins the match. Each game is won by Fischer with probability p, by Spassky with probability q, and is a draw with probability 1-p-q . • What is the probability that Fischer wins the match? • What is the PMF, the mean, and the variance of the duration of the match? Problem 4 You are visiting the rainforest, but unfortunately your insect repellent has run out. As a result, at each second, a mosquito lands on your neck with probability 0.4. If a mosquito lands, it will bite you with probability 0.4, and it will never bother you with probability 0.6, independently of other mosquitoes. What is the expected time between successive bites? Problem 5 Let X 1 ,...,X n be independent, identically distributed random variables with common mean and variance. Find the values of c and d that will make the following formula true: E [( X 1 + .... + X n ) 2 ] = cE [ X 2 1 ] + d ( E [ X 1 ]) 2 ....
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