Quiz2-2007-2- test 2- 610 question 1

Let y g x x 2 then i y 0 and fy y fx

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Unformatted text preview: y increasing with respect to X . Let Y = g (X ) = X 2 . Then, I (y ) = (0, ∞), and fY (y ) = fX g −1 (y ) · d −1 g (y ) dy • g − 1 ( y ) = y 1 /2 . • • d −1 (y ) dy g = 1 y −1/2 . 2 d −1 (y ) dy g = 1 y −1/2 . 2 • fX g −1 (y ) = λe−λy 1/2 . • So, fY (y ) = λ −1/2 −λy1/2 y e , 2 3. Consider the joint probability density function k (1 + 2xy ) fX,Y (x, y ) = 0 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 otherwise for the jointly distributed continuous random variables X and Y . 1 y > 0. , (a) (4%) Find the value of k . Sol) 1 1 1=k (1 + 2xy )dxdy = 0 0 2 3 k =⇒ k = 2 3 (b) (4%) Find the marginal probability density function of Y , fY (y ). Sol) 2 3 1 (1 + 2xy )dx = 0 fY (y ) = 22 +y 33 2 + 2y 0≤y≤1 0 otherwise 3 3 (c) (4%) Find the conditional probability density function of X given that Y = y , fX |Y (x|y ). Sol) fX |Y (x|y ) = = = fX,Y (x,...
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This document was uploaded on 11/01/2013.

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