Problems in Probability

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Fall 2011 Strategic Practice 7: Section 1 (Joint, Conditional, and Marginal Distributions) - Question 3
Let be i.i.d. Unif(0, 1), and let L = min() and M = max().
(a) Find the marginal CDF and marginal PDF of M, and the joint CDF and joint PDF of L, M. Hint: for the latter, start by considering .
(b) Find the conditional PDF of M given L.
Solution: (a) For m > = l: CDF of M is m^3, PDF of M is 3m^2. Joint CDF of L and M is m^3 - (m - l)^3, Joint PDF of L and M is 6(m - l). (b) 2(m - l)/(1 - l)^2 for m and l between 0 and 1 inclusive, and m >= l.
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