Exercise 7, Ch 7 (the stick problem)

The problem

Directly copied description from the book (and thus the text is often scrambled), Blitzstein and Jessica Hwang, Introduction to Probability, Second Edition:

A stick of length L (a positive constant) is broken at a uniformly random point X.
Given that X = x, another breakpoint Y is chosen uniformly on the interval [0; x].
(a) Find the joint PDF of X and Y . Be sure to specify the support.
(b) We already know that the marginal distribution of X is Unif(0;L). Check that
marginalizing out Y from the joint PDF agrees that this is the marginal distribution of
X.
(c) We already know that the conditional distribution of Y given X = x is Unif(0; x).
Check that using the de_nition of conditional PDFs (in terms of joint and marginal
PDFs) agrees that this is the conditional distribution of Y given X = x.
(d) Find the marginal PDF of Y .
(e) Find the conditional PDF of X given Y = y.

Tags
Mathematical statistics, probability, uniform distribution, joint distribution, marginal distribution, and conditional distribution.

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