HW11 ConvolutionPractice_ans_1 - Practice Problems:...

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Unformatted text preview: Practice Problems: Computing the density of X + Y . 1) Let X ∼ Exp(λ) and Y ∼ Exp(µ), and let X and Y be independent. Compute the p.d.f. of X + Y . 1 2) Let X and Y be independent with marginal density functions ￿ ￿ 2x 0 ≤ x ≤ 1 3/4(1 − y 2 ) −1 ≤ y ≤ 1 fX (x) = and fY (y ) = . 0 otherwise 0 otherwise Compute the density of X + Y . 2 3) Let X and Y be independent random variables such that X ∼ Exp(2) and Y ∼ U (−1, 1). Compute the p.d.f. of X + Y . 3 4) Let (X, Y ) have joint p.d.f. ￿ x+y f (x, y ) = 0 a) Are X and Y independent? b) Compute the p.d.f. of Z = X + Y . 4 0 ≤ x, y ≤ 1 . otherwise 5) Let X and Y be independent random variables with X ∼ Exp(λ) and Y ∼ Exp(λ). Compute the p.d.f. of X − Y . There are two ways to approach this problem. One way is to view Z = −Y as a random variable and then use convolutions to compute the density of X + Z . The second way to do this problem is to use the “general method.” That is, calculate P (X − Y ≤ t) and then take the derivative with respect to t. V-U 5 ...
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