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.