Assignment_2

# Assignment_2 - Y(c the conditional p.d.f of X given Y = y(d...

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1 Solutions of Assignment #2 - STAT 330 Due in class: Thursday Feb. 25 Important Note: You need to print out this page as the cover page for your assignment. LAST NAME: FIRST NAME: ID. NO.: QUESTION 1. /8 QUESTION 2. /6 QUESTION 3. /4 QUESTION 4. /12 QUESTION 5. /6 QUESTION 6. /4 TOTAL: /40

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2 1. Suppose X and Y have joint p.d.f f ( x,y ) = 6 x 2 y, 0 < x < 1 , 0 < y < 1, and zero otherwise. Compute (1). P ( X 2 <Y ) (2). P (0 . 5 <X + Y< 1 . 5) (3). P ( Y> 0 . 5 | X> 0 . 5) (4). P ( Y> 0 . 6 | X =0 . 6)
3 2. Let X and Y have the joint density f ( x,y )= e - y , 0 x y (a) Find COV ( X,Y ) and correlation of X and Y (b) Find E ( X | Y = y ) and E ( Y | X = x ) (c) Find the density functions of the random variables E ( X | Y ) and E ( Y | X )

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4 3. Let f ( x,y )= c ( x 2 - y 2 ) e - x , 0 x< , - x y < x . (a). Find c. (b). Find the marginal densities.
5 4. Suppose X and Y are continuous r.v.’s. Suppose also that X has the p.d.f. f X ( x )= 5 8 (1 - x 4 ) , - 1 < x < 1 and the conditional p.d.f. of Y given X = x is f Y | X ( y | x )= k ( x 2 + y ) , - 1 < x < 1 , 0 < y < 1 - x 2 Determine (a). k (b). the marginal p.d.f. of

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Unformatted text preview: Y (c). the conditional p.d.f of X given Y = y (d). E ( Y | X ) and E ( X | Y ) (e). Cov ( X,Y ) (f). P ( Y ≤ X + 1) (g). Are X and Y independent ? 6 5. For each of the following questions, ±nd E ( X ) and V ar ( X ). (a). Suppose X | Y = y ∼ POI( y ), and Y ∼ GAM( α,β ) with the pdf f ( y ) = 1 Γ( α ) β α y α-1 e-y/β , for y > . (b). Suppose X | Y = y ∼ BIN( y,p ) and Y ∼ POI( λ ). (c). Suppose X | Y = y ∼ BIN( y + 1 ,p 1 ) and Y ∼ BIN( n,p 2 ). 7 6. Suppose X and Y have the joint pdf f ( x,y ) = e-x-y , x > , y > , and zero otherwise. (a). Find P ( X + Y < 1). (b). Let U = X + Y , and V = X . Find the joint pdf of U and V ....
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Assignment_2 - Y(c the conditional p.d.f of X given Y = y(d...

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