1. Suppose U~ Uniform(0, 1), X=-alog(U) for a > 0, and Y = WX where W...

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1. Suppose U~ Uniform(0, 1), X=-alog(U) for a > 0, and Y = WX where W is discrete random variable with P(W=-4) = P(W = 4) = and is independent of X. (a) Derive the cdf of X and identify its distribution. (b) Derive the distribution of Y and rigorously verify. (c) Compute .05 quantiles of: (i) Y when a = 2 (ii) (U-1) (iii) (U-4). 1. Suppose U~ Uniform(0, 1), X=-alog(U) for a > 0, and Y = WX where W is discrete random variable with P(W=-4) = P(W = 4) = and is independent of X. (a) Derive the cdf of X and identify its distribution. (b) Derive the distribution of Y and rigorously verify. (c) Compute .05 quantiles of: (i) Y when a = 2 (ii) (U-1) (iii) (U-4).