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160a09_mid - Name PSTAT 160A Winter 09 Midterm Exam Problem...

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Name: PSTAT 160A Winter 09 Midterm Exam Problem 1. (30 points) In this problem, we want to simulate the random variable X with pdf given by p X ( k ) = p 2 k (1 - p ) k - 1 , k = 1 , 2 , · · · 1) Show that this is a valid pdf (hint: the mean of a Geometric( p ) is 1 /p ) 2) Give the simple algorithm seen in class to simulate from a Geometric distribution 3) We want to simulate X with rejection method, using a Geometric ( p 0 ) random variable Y . Set-up the method in this example. Give the values of p and p 0 that can work. Hint: the function xr x , x 0 , 0 < r < 1 fixed, has a maximum at x o = - 1 /log ( r ). 4) What is the expected number of steps required to generate a value for X ? 5) What should be the optimal choice for p 0 ? (you should work on this question once you’re done with the rest of your exam)
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2 Problem 2. (30 points) (very short, very simple questions!!!!) Let ( X, Y ) have joint pdf such that the points (0 , 0) , (0 , 1) , (1 , 0) have the same probability. a) Are X and Y independent and why?
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