ECE6303 HW 6 Fall 2018 1. X and Y have joint density function f ( x, y) XY 1 xy A , | x | 1,| y | 1; zero,otherwise. (1) Find A so that the above defined is a valid joint density function. (2) Are X,Y uncorrelated? Show your answer. (3) Are X,Y independent? Show your answer. (4) Find the joint density function of X 2 and Y 2 (5) Prove or disprove that X 2 and Y 2 are independent. (Video is required) 2. f ( x, y) B, x2 y 2 1; zero, otherwise. XY (a) Find B so that the above defined function is a valid joint pdf. (b) Find E{X }, E{X | y}, E{X |Y}, E{Y}, E{Y | x}, E{Y | X } and E{XY} . (c) Prove or disprove X and Y are uncorrelated. (Video is required) 1 3. X and Y are i.i.d. random variables with common p.d.f. f (x) exu(x), f (y) eyu(y). X Y f X ( x) f Y ( y) x y (a) Z X Y . Find fZ ( z) by characteristic function method. (b) Z XY . Find fZ ( z) by auxiliary variable method. 4. X and Y are independent with exponential densities f X ( x) e xu( x), fY ( y) e yu( y). Find the joint density Z=X+Y and W=X/Y. 5. Suppose the density function of X is an even function. Prove or disprove that X and |X| are uncorrelated. Prove or disprove that X and |X| are independent. 6. Suppose X and Y are independent normal random variables. Let 2 Z X Y (where , are nonzero constants) W X Y Given 2 x2 2 y2 . Prove or disprove that Z and W are independent normal random variables. 7. Prove that for any X, Y real or complex (a) | E{ XY }|2 E{| X |2}E{| Y |2} (b) (triangle inequality) E{| X Y |2} E{| X |2} E{|Y |2}. 8. X , Y and Z are i.i.d. random variables uniform in (0, 1). Find E{ X }. X Y Z 9. Let X and Y be i.i.d. random variables with common probability mass function P( X k ) P(Y k ) pqk , k 0,1,2,... Define Z min{X ,Y }. Find the probability mass function of Z. 3 Axy, 0 x y 1, 0, otherwise. 10. X and Y have joint density function f ( x, y) XY (a) Find A. (b) Find and sketch the marginal density functions f ( x) and f ( y) . (c) Prove or disprove that X,Y are independent. (d) Find E{X }, E{Y} and E{XY } . (e) Prove or disprove that X,Y are uncorrelated. (f) Find and sketch the conditional density functions f ( y | x) and f ( x | y) . (g) Find E{Y | x}, E{Y | X } and E{X | y}, E{X |Y}. X Y Ay 2 , 0 x| y|1 11. Given the joint probability density function f XY ( x, y) 0, otherwise. 4 1 y fXY ( x, y) yx 1 x x y x 1 y 1 1 (a) Find A. Find E{X }, E{Y} and E{XY } . (b) Prove or disprove that X,Y are uncorrelated and/or independent. (c) Find and sketch the conditional density functions f ( y | x) and f ( x | y) . (d) Find E{Y | x}, E{Y | X } and E{X | y}, E{X |Y}. Sketch E{X | y}. 5