A one-foot-long stick is broken at a point X (measured from the left end) chosen randomly uniformly along its length. Then the left part is broken at a point Y chosen randomly uniformly along its length. In other words, X is uniformly distributed between 0 and 1 and, given X = x, Y is uniformly distributed between 0 and x.

a. Determine E(Y|X = x) andthenV(Y|X = x).
b. Determine ƒ(x,y) using ƒX(x) and ƒY|X(y|x).

c. Determine ƒ_Y(y).
d. Use ƒ_Y(y) from (c) to get E(Y) and V(Y).
e. Use (a) and the Laws of Total Expectation and Variance to get E(Y) and V(Y).