Question

Question 4 [16 marks] X Y (a) The random vector has probability density function fx.y (x, y)exp {-22 - 2xy - 3y*}, where k is some constant. (i) Find k N (0, 3/2) and Y ~ N (0,1/2) 11 Show that X Find cov (X, Y) and corr (X, Y) 111 (iv) Find E (Y|X) (b) The random variables U and V are distributed with mean 1/A, while V is uniformly distributed between 0 and 1 independent. U is exponentially (i) Show, using the convolution formula, that the pdf of W = U+ V 1 - e-Aw e-A(w-1) 0 w1 -Aw fw (w) w 1 otherwise (ii) Show that the cdf Fw (w) given by 15 w0 1 (1-w) 0 w1 Fw (w) 22 - 1-) -A (w-1) w> 1 and hence verify that fw (w) is a valid pdf.

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Answer 1

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