hw7 - variables that are uniformly distributed between[0 1...

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EE 351K Probability and Random Processes FALL 2010 Instructor: Haris Vikalo [email protected] 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 Chebyshev’s inequality and assuming x ≥ n 2 , a) Provide a bound on Pr (Σ n i =1 X i > x ) . b) Use the central limit theorem approximation to provide an approximation (in terms of the φ function) for the expression in (a). In other words, provide an expression for Pr (Σ 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 ( θ ) = E [ e θX ] . a) Show that for any θ ≥ , Pr ( e θX ≥ e θa ) ≤ E [ e θX ] e θa . b) Argue that Pr ( X ≥ a ) = Pr ( e θX ≥ e θa ) and that, therefore, Pr ( X ≥ a ) ≤ e-[ θa-log M X ( θ )] ....
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This note was uploaded on 04/10/2011 for the course EE 351k taught by Professor Bard during the Spring '07 term at University of Texas.

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