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Statistics / Introduction to Probability, 2 / Chapter 3 / Problem 21

Introduction to Probability, | 2nd Edition | ISBN: 9781886529236 | Authors: Dimitri P. Bertsekas John N. Tsitsiklis

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We start with a stick of length e. We break it at a point

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QUESTION:

We start with a stick of length e. We break it at a point which is chosen according to a uniform distribution and keep the piece, of length Y, that contains the left end of the stick. We then repeat the same process on the piece that we were left with, and let X be the length of the remaining piece after breaking for the second time. (a) Find the joint PDF of Y and X. (b) Find the marginal PDF of X. (c) Use the PDF of X to evaluate E[X]. (d) Evaluate E[X] , by exploiting the relation X = y. (X/ Y).

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QUESTION:

We start with a stick of length e. We break it at a point which is chosen according to a uniform distribution and keep the piece, of length Y, that contains the left end of the stick. We then repeat the same process on the piece that we were left with, and let X be the length of the remaining piece after breaking for the second time. (a) Find the joint PDF of Y and X. (b) Find the marginal PDF of X. (c) Use the PDF of X to evaluate E[X]. (d) Evaluate E[X] , by exploiting the relation X = y. (X/ Y).

ANSWER:

Step 1 of 4

(a) We have , for . Furthermore, given the value of , the random variable X is uniform in the interval .

Therefore, , for .

We conclude that

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