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Assume that we start with a stick of length L. Then we break it at a...

Assume that we start with a stick of length L. Then we break it at a point is chosen randomly and uniformly over its length, and keep the piece that contains the left end of the stick. We then repeat the same process on the stick we were left with. What is the expected length of the stick that we were left with, after breaking it twice?

Math   Statistics-And-Probability   
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Answer #1

As we are keeping the left piece of a uniformly cut piece, therefore the PDF for the remaining part here is obtained as:

Give the length X as the remaining length after first cut, the probability density of the piece left after second cut is given as:

Therefore the expected length of the stick that we are left with after breaking it twice is computed using the Tower law here as:

Therefore L /4 = 0.25L is the required expected value of the stick after breaking it twice.

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