{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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,...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online