# hw6 - mean(a Show that X and E X | Y are positively...

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EE 351K Probability, Statistics, and Random Processes SPRING 2009 Instructor: Shakkottai/Vishwanath { shakkott,sriram } @ece.utexas.edu Homework 6 (Due 04/01/2009) Problem 1 Find the MGF associated with the random variable X with pmf p X ( k ) = ± p e - λ λ k k ! + (1 - p ) e - μ μ k k ! if k = 0 , 1 ,..., 0 otherwise where λ and μ are positive scalars, and p satisﬁes 0 p 1 . Problem 2 The MGF associated and the mean associated with a discrete random variable X are given by M X ( θ ) = ae θ + be 2( e θ - 1) , E [ X ] = 5 Determine: (a) The scalar parameters a and b . (b) p X (1) , E [ X 2 ] , and E [2 X ] . Problem 3 Let X be a continuous uniform random variable on [0 , 1] and let Y be a continuous uniform random variable on [7 , 10] . Assume that X and Y are independent. Determine an expression and sketch the pdf of Z = X + Y . Problem 4 Let X be a geometric random variable with parameter P , where P is itself random and uniformly distributed on [0 , ( n - 1) /n ] , where n is a positive integer. In other words, for each k = 1 , 2 ,..., Pr ( X = k | P = p ) = (1 - p ) k - 1 p. Let Z = E [ X | P ] . Find E [ Z ] and lim n →∞ E [ Z ] . Problem 5 Consider two random variables X and Y . Assume for simplicity that they both have zero
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Unformatted text preview: mean. (a) Show that X and E [ X | Y ] are positively correlated. (b) Show that the correlateion coefﬁcient of Y and E [ X | Y ] has the same sign as the correlation coefﬁcient of X and Y . Problem 6 Let X,Y,Z be independent uniformly distributed random variables on [0 , 1] . Let W = max( X,Y,Z ) and R = min( X,Y,Z ) . Find the pdf of W , the pdf of R , and the joint pdf of ( W,R ) . Problem 7 We are given that E [ X ] = 2 , E [ Y ] = 3 , E [ X 2 ] = 5 , E [ Y 2 ] = 12 and E [ XY ] = 1 . Find the linear least squares estimator of Y given X . Problem 8 Consider three zero mean random variables X,Y,Z with known variances and covari-ances. Give a formula for the linear least squares estimator of X given Y and Z , that is, ﬁnd a and b that minimize E [( X-aY-bZ ) 2 ] . For simplicity, assume that Y and Z are uncorrelated....
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