1 suppose that x y and z have a joint density function, Ask an Expert

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1. Suppose that X, Y , and Z have a joint density function given by (e-(x+y+z), for x, y, z > 0, f (x, y, z) = 0, otherwise. Compute P (X < y =""><>Z) and P (X = Y Z).

2. Two points are chosen uniformly and independently on the perimeter of a circle of radius 1. This divides the perimeter into two pieces.

Determine the expected value of the length of the shorter piece.

3. Let X1 and X2 be independent, U (0, 1)-distributed random variables, and let Y denote the point that is closest to an endpoint. Determine the dis- tribution of Y.

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