homework8 - of & is equal to the length of the...

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C OMER ECE 600 Homework 8 1. Show that if g( x ) = E[Y|X= x ] is the minimum mean square error estimate of Y in terms of X, then (X)] E[ ] [Y ] (X)) E[(Y 2 2 2 g E g - = - 2. Let X 1 ,…,X n be random variables that are independent with finite variances. Form i n i i X Y 1 = = α where i is real for every i . Find the mean and variance of Y. 3. Show that if R is the correlation matrix of the random vector X = (X 1 ,…,X n ) and R -1 is its inverse, then E[X R -1 X T ] = n . 4. Let X 1 , X 2 ,…, X n be n i.i.d random variables having uniform distribution on [0,1]. Also, let X (1) and X ( n ) be random variables denoting the minimum and maximum of the X i ’s, respectively. We define the random variable Y = X (1) + X ( n ) . Compute the pdf of Y. 5. Let be selected from the interval = [0,1], where the probability that ± is in a subinterval
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Unformatted text preview: of & is equal to the length of the subinterval. For n 1, define the following random sequences: a. n n = ) ( U b. -= n n 1 1 ) ( V c. n n e ) ( W = d. n n 2 cos ) ( Y = e. ) 1 ( e ) ( Z--= n n n Which of these sequences converges everywhere? almost everywhere? Identify the limiting random variable. 6. Let X n and Y n be two (possibly dependent) sequences of random variables that converge in the mean square sense to X and Y, respectively. Does the sequence X n + Y n converge in the mean square sense, and, if so, to what limit? 7. Show that if a n a and E[|X n a n | 2 ] 0, then X n a in the mean square sense as n ....
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