hw6sol - Stat 309 1/2 Quoc Tran HW 6 Solutions 7 Find the...

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Unformatted text preview: Stat 309 1/2 Quoc Tran HW 6 Solutions 7. Find the joint and marginal densities corresponding to the cdf: F x , y = 1 − e − x 1 − e − y , x 0, y 0, 0, Ans: Joint density (joint pdf, jpdf): f X ,Y x , y = d 2 dxdy F X , Y x , y = d 2 dxdy 1 − e − x 1 − e − y = d dx 1 − e − x e − y = e − x e − y Marginal densities: Marginal density of X: f X x = ∫ ∞ f X ,Y x , y dy = ∫ ∞ e − x e − y dy = e − x ∫ ∞ e − y dy = e − x Marginal density of Y: f Y y = ∫ ∞ f X ,Y x , y dx = ∫ ∞ e − x e − y dx = e − y ∫ ∞ e − x dx = e − y 14. Suppose that f x , y = x e − x y 1 , x ∞ , y ∞ a. Find the marginal densities of X and Y. Are X and Y independent? b. Find the conditional densities of X and Y. Ans: a. Marginal densities: Marginal density of X: f X x = ∫ ∞ f x , y dy = ∫ ∞ xe − x y 1 dy = ∫ ∞ x e − xy e − x dy = e − x ∫ ∞ x e − xy dy = e − x Marginal density of Y: f Y...
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hw6sol - Stat 309 1/2 Quoc Tran HW 6 Solutions 7 Find the...

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