Question: solve We recall the Gamma function given by T(t) = Jo at le adx, t 0. t-1 e if x 0 Let a r.v. X

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We recall the Gamma function given by T(t) = Jo at le adx, t 0. t-1 e if x 0 Let a r.v. X with Gamma(1, t) distribution: f (x) = r(t) otherwise. 1) Let the Laplace transform of X be the function A " ()) = fore fx(x)dx, 1 20. Show that x ()) = (142) 2) Let a -t. Give the expression of E[X"] = Spa*fx(x)dx. 3) We admit that the Laplace transform entirely characterize the distribution of a positive r.v. What is the distribution of X + Y , where Y has the Gamma(1, s) distribution, s 0

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