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Assign1 - Introduction to Stochastic Processes Spring 2011...

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Introduction to Stochastic Processes, Spring 2011 Homework #1 Due: Thursday, January 13th, 2011 in class 1. Recall that a geometric random variable is a discrete one taking values in the set { 1 , 2 , 3 , . . . , } , with P ( X = k ) = p (1 - p ) k - 1 where 0 p 1. (i) The random number X should always be thought of as the number of independent trials it takes before an experiment succeeds, where p is the probability of success on any given trial. Give a brief explanation of why this is from the formula for P ( X = k ). (ii) Compute the mean of X . (iii) Compute the variance of X . 2. A bowl contains twenty cherries, exactly fifteen of which have had their stones removed. A greedy pig eats five whole cherries, picked at random, without remarking on the presence or absence of stones. Subsequently, a cherry is picked at random from the fifteen. (i) What is the probability that the cherry contains a stone? (ii) Given that this cherry contains a stone, what is the probability that the pig consumed at least one stone?
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