Problems in Probability

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Fall 2011 Homework 8: Question 5
Let X and Y be independent positive r.v.s, with PDFs and respectively, and consider the product T = XY . When asked to find the PDF of T, Jacobno argues that "it's like a convolution, with a product instead of a sum. To have T = t we need X = x and Y = t/x for some x; that has probability , so summing up these possibilities we get that the PDF of T is ." Evaluate Jacobno's argument, while getting the PDF of T (as an integral) in 2 ways: (a) using the continuous law of total probability to get the CDF, and then taking the derivative (you can assume that swapping the derivative and integral is valid); (b) by taking the log of both sides of T = XY and doing a convolution (and then converting back to get the PDF of T).
Solution: (a) the transformation (X, Y) to (XY, X) is nonlinear, in contrast to the transformation (X, Y) to (X + Y, X) considered in SP 8 # 2.3. Jacobno is ignoring the distinction between probabilities and probability densities, and is implicitly (and incorrectly) assuming that there is no Jacobian term. (b) This concurs with (a): Jacobno is missing the x in the denominator. See complete solution at iTunes.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."
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