homework7 - EL 630: Homework 7 1. X ∼ P(λ ); ( X is...

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Unformatted text preview: EL 630: Homework 7 1. X ∼ P(λ ); ( X is Poisson with parameter λ ), i.e., P(X = k ) = e − λ λk , k = 0,1,2, k! . Find E 1 . 1 + X 2. Let X and Y be jointly distributed with p.d.f 1 + xy , | x |< 1, | y |< 1, f XY ( x, y ) = 4 0, otherwise . (a) Find f X ( x ), f Y ( y ). (b) Are X and Y independent? 3. Let X, Y have the joint p.d.f defined by 3 xy + (x 2 / 2 ) , 0 < x < 1, 0 < y < 2, f XY ( x, y ) = 4 otherwise. 0, [ (a) Find f Y ( y ). (b) Determine P(Y < 1 | X < 1 / 2 ). 4. Let X, Y be jointly distributed with p.d.f 1 1 + xy (x 2 − y 2 ) , | x |≤ 1, | y |≤ 1, f XY ( x, y ) = 4 otherwise. 0, [ (a) Compute the marginal p.d.fs of X and Y. (b) Are X and Y independent? 1 5. X and Y have the joint distribution xy , x > 0, y > 0, FXY ( x, y ) = (1 + x )(1 + y ) otherwise. 0, Find the marginal p.d.fs of X and Y. Are X and Y independent? 6. X and Y are bivariate (jointly) Normal r.vs with joint p.d.f x2 xy y 2 1 1 2 − 2ρ f XY ( x, y ) = + 2 , − ∞ < x < ∞, − ∞ < y < ∞. exp − σ 1σ 2 σ 2 2πσ 1σ 2 (1 − ρ 2 )1 / 2 2(1 − ρ 2 ) σ 1 and | ρ |< 1. Find f X ( x ) and f Y ( y ). Show all steps. (Hint: Use substitutions that were used in deriving Φ X (u ) for a Normal r.v in class). 2 ...
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homework7 - EL 630: Homework 7 1. X ∼ P(λ ); ( X is...

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