- #1

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## Main Question or Discussion Point

This is from a chapter on distributions of two random variables. Let X and Y have the pdf f(x,y) = 1, 0<x<1 and 0<y<1, zero elsewhere. Find the cdf and pdf of the product Z=XY.

My current approach has been to plug in X=Z/Y in the cdf P(X<=x) , thus P(Z/Y<=x), and integrate over all values of Y. The integral I've tried to solve is ∫(0 to 1)∫(0 to Z/Y) 1 dxdy (sorry for the formatting).

This is somehow wrong, because from that integral I get z*(ln(1) - ln(0)), which is undefined (unless I'm solving the integral incorrectly, which would actually be a relief).

The correct answer is -ln(z), 0<z<1, though it's amiguous as to whether that's the cdf or pdf.

Thanks for any solutions or suggestions about where my approach is going wrong.

My current approach has been to plug in X=Z/Y in the cdf P(X<=x) , thus P(Z/Y<=x), and integrate over all values of Y. The integral I've tried to solve is ∫(0 to 1)∫(0 to Z/Y) 1 dxdy (sorry for the formatting).

This is somehow wrong, because from that integral I get z*(ln(1) - ln(0)), which is undefined (unless I'm solving the integral incorrectly, which would actually be a relief).

The correct answer is -ln(z), 0<z<1, though it's amiguous as to whether that's the cdf or pdf.

Thanks for any solutions or suggestions about where my approach is going wrong.