10s418hw5sol

# 10s418hw5sol - α 1 and(n s Therefore the conditional...

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1 STAT 418 HW# 5 Solution 6.2 a) 8 7 (0,0) = 13 12 8 5 (0,1) = 13 12 5 4 (1,1) = 13 12 P P P × × × × × × b) 8 7 6 (0,0,0) = 13 12 11 8 7 5 (0,0,1) = (0,1,0) = (1,0,0) = 13 12 11 8 5 4 (0,1,1) = (1,0,1) = (1,1,0) = 13 12 11 5 4 3 (1,1,1) = 13 12 11 P P P P P P P P × × × × × × × × × × × × × × × × 6.10 a) - ( ) e x X f x = , - ( ) e y Y f y = , and , 0 x y > Therefore 1 ( ) 2 P X Y < = b) - ( ) 1- a P X a e < = 6.20 a) - - Yes, ( ) ( ) ; 0 , and 0 x y X Y f x xe and f y e x y = = < < ∞ < < ∞ b) No, 1 ( ) ( , ) 2(1 ); 0 1 X x f x f x y dy x x = = - < < 0 ( ) ( , ) 2 ; 0 1 y Y f y f x y dx y y = = < <

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2 6.21 a) We must show that ( , ) 1 f x y dxdy ∞ ∞ -∞ -∞ = ∫ ∫ , 1 1 0 0 1 2 0 1 2 3 0 ( , ) 24 12 (1 ) 12( 2 ) 1 1 2 12( ) 2 4 3 1 y f x y dxdy xydxdy y y dy y y y dy - ∞ ∞ -∞ -∞ = = - = - + = + - = ∫ ∫ ∫ ∫ b) 1 0 1 1 0 0 1 2 2 0 ( ) ( ) 24 12 (1 ) 2/5 X x E X xf x dx x xydydx x x dx - = = = - = ∫ ∫ c) 2/5 6.27 1 2 1 2 1 2 0 0 2 0 1 1 2 1 1 2 ( ) (1 ) therefore; ( ) ay x y ay y X P a x e e dxdy Y e e dy a a X P a Y λ - - - - < = = - = + < = + ∫ ∫
3 6.32 a) 2 e - b) 2 2 2 1 2 1 3 e e e - - - - - = - 6.35 a) 1 2 1 2 5 ( 1 | 1) 13 8 ( 0 | 1) 13 P X X P X X = = = = = = b) same as in (a) 6.43 1 1 ( 1) 1 2 ( | ) ( ) ( | ) ( ) = ( ) = n a s a n s P N n g f n P N n c e ae a c e λ - - - - + + - = = = Where C 1 and C 2 don’t depend on . But from the preceding we can conclude that the conditional density is gamma with parameters (

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Unformatted text preview: α +1) and (n+s). Therefore the conditional expectation equal to (n+s)/( +1). 6.55 a) If u xy = , x v y = then 2 1 y x J x y y = - = 2 x y - and u y v = , x uv = . b) , , 2 1 1 ( , ) ( , ) 2 2 U V X Y u f u v f uv v v vu = × = ; 1 1, u v u u ≥ < < and 2 2 1/ 1 1 ( ) log 2 u U u f u dv u vu u = = ∫ ; 1 u ≥ For 1 v > , 2 2 1 1 ( ) 2 2 V v f v du vu v ∞ = = ∫ ; 1 v > For 1 v < , 2 1/ 2 1 1 ( ) 2 2 V f v du vu ∞ = = ∫ ; 0 1 v < < 4 Theoretical 9 1 1 2 (min{ ,..., } ) ( , ,..., ) are independent ... n n i t t n t P X X t P X t X t X t X e e e λ---> = > > > = × × = Q...
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10s418hw5sol - α 1 and(n s Therefore the conditional...

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