hw7 - variables that are uniformly distributed between [0 ,...

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EE 351K Probability and Random Processes FALL 2010 Instructor: Haris Vikalo hvikalo@ece.utexas.edu Homework 7 Due on : Tuesday 10/26/10 Problem 1 Let F be a non-decreasing function with lim x →-∞ F ( x ) = 0 and lim x + F ( x ) = 1 . Let U be a uniform random variable on [0 , 1] . 1. Let X = F - 1 ( U ) . What is the CDF of X ? (Note F - 1 is the inverse of F . A function g is the inverse of F if F ( g ( x )) = x for all x .) 2. How can you generate an exponential random variable from U ? Problem 2 Let X and Y be two random variables that are independent and uniformly distributed over the interval [0 , 2] . Find the CDF and PDF of | X - Y | . Problem 3 Consider two random variables X and Y . For simplicity, assume that they both have zero mean. (a) Show that X and E [ X | Y ] are positively correlated. (b) Show that cov ( Y,E [ X | Y ]) = cov ( X,Y ) . Problem 4 Let P denote a continuous random variable uniformly distributed on [0 , n - 1 n ] , where n is a positive integer. Let X denote a discrete random variable such that Pr ( X = k | P = p ) = p k - 1 (1 - p ) . Let Z = E [ X | P ] . Find E [ Z ] and lim n →∞ E [ Z ] . Problem 5 Let X i ,i = 1 , 2 ,... be independent identically distributed (i.i.d.) continuous random
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Unformatted text preview: variables that are uniformly distributed between [0 , 1] . Using Chebyshevs inequality and assuming x n 2 , a) Provide a bound on Pr ( n i =1 X i > x ) . b) Use the central limit theorem approximation to provide an approximation (in terms of the function) for the expression in (a). In other words, provide an expression for Pr ( n i =1 X i > x ) using the central limit theorem. c) Compute (a) and (b) for n = 50 , and x = 80 and comment on your answers. Problem 6 Let X i ,i = 1 , 2 ,...,n be n i.i.d. random variables, with M x ( ) = E [ e X ] . a) Show that for any , Pr ( e X e a ) E [ e X ] e a . b) Argue that Pr ( X a ) = Pr ( e X e a ) and that, therefore, Pr ( X a ) e-[ a-log M X ( )] ....
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