t209soln - STAT 330 TEST 2 SOLUTIONS 1. Suppose X and Y are...

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STAT 330 TEST 2 SOLUTIONS 1 . Suppose X and Y are random variables. Suppose also that Y has p.d.f. f 2 ( y )= e y ,y > 0 and the conditional p.d.f. of X given Y = y is f 1 ( x | y e x + y , 0 <y<x< . ( a )[3 ] Find the marginal p.d.f. of X . The marginal p.d.f. of X is f 1 ( x Z −∞ f ( x,y ) dy = Z −∞ f 1 ( x | y ) f 2 ( y ) dy by the Product Rule = x Z 0 e x + y e y dy = e x x Z 0 dy = xe x ,x > 0 Note: X v GAM (2 , 1) . ( b )[2 ] Find the conditional p.d.f. of Y given X = x . The conditional p.d.f. of Y given X = x is f 2 ( y | x f ( ) f 1 ( x ) = e x xe x = 1 x , 0 . Note: Y | X = x v UNIF (0 ) . 1
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2 . Let P and Y be random variables such that P has p.d.f. f 1 ( p )= α p α 1 , 0 <p< 1; α > 0 and Y | P = p v BIN ( n,p ) . Note: If X BIN ( n, θ ) then E ( X n θ and Var ( X n θ (1 θ ) . ( a )[1 ] Show E ¡ P k ¢ = α α + k ,k 0 . E ¡ P k ¢ = Z 1 0 p k α p α 1 dp = α p α + k α + k | 1 0 ¸ = α α + k . ( b )[ 5 ] Find E ( Y ) and ( Y ) . You need not simplify the expression for ( Y ) . E ( Y E [ E ( Y | P )] = E ( nP ) since Y | P = p v BIN ( ) = nE ( P n μ α α +1 .
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t209soln - STAT 330 TEST 2 SOLUTIONS 1. Suppose X and Y are...

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