(Answered) Question a stick of length l a positive constant is broken

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Question: 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 definition 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.

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