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1. * (Uniform Distribution in a Sphere). Suppose X; Y; Z are uni- formly distributed in the unit sphere. Find the mean and the variance of the distance of the point .X; Y; Z/ from the origin.

2. * (Uniform Distribution in a Sphere). Suppose X; Y; Z are uni- formly distributed in the unit sphere.

(a) Find the marginal density of .X; Y /.

(b) Find the marginal density of X.

3. Suppose X; Y; Z are independent exponentials with means ?; 2?; 3?. Find P.X Y Z/:

4. (Mean Residual Life). Suppose X "" N.µ; a 2/. Derive a for- mula for the mean residual life and investigate its monotonicity behavior with respect to each of a; c; µ, each time holding the other two fixed.

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