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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- Spring '07
- Probability theory, 351K, Random Processes Instructor, email@example.com, 0 - k