Question: Let U, X be independent random variables, where U is Uniform (0, 1) distributed and X is Exponential () distributed. Note that U, X have

Let U, X be independent random variables, where U is Uniform (0, 1) distributed and X is Exponential () distributed. Note that U, X have the joint density fU (u)fX(x) = { exp(x), for 0 u 1, x 0, 0, otherwise. Let c > 0 be a constant. Find the conditional density of X, given the event U exp(cX).

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