Use the law of iterated expectation to calculate
Suppose we have a stick of length L. We break it once at some point X _ Unif(0;L). Then we break it again at some point Y _ Unif(0;X). Use the law of iterated expectation to calculate E[Y ].
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X is the length of the stick after we break for the ?rst time. Y is the length after the second time.
We have E[ Y | X ] = X /2, since the breakpoint is chosen uniformly over the length X of the remaining stick. similarly, E[X ] = L/2.
E[Y] = E[E [Y | X ] ]= E[X/2]=E[X]/2 = L/4
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