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Let $X$ and $Y$ be independent ,uniform over [0,1].

I need to find the pdf of XY. So let $U = X, V= XY$ and Jacobian $J$ can be calculated as $J= 1/u$. So $$f_{U,V}(u,v) = f_{X,Y}(x,y)|J|= 1/u$$ So we get $f_V(v) = \int_0^1 \frac 1u du $ which does not exist.

Can anyone tell me where I am going wrong?

The answer in text book is different than what I am getting.

quantum_spin
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    Your change of variables formula is incorrect. You need to keep track of the support of the distributions, without which densities don't make sense. Here it is easier to work out the cdf (draw a picture) and differentiate to get the pdf. – StubbornAtom Nov 24 '21 at 13:21
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    Useful techniques are also illustrated at https://stats.stackexchange.com/questions/473325 for a very closely related question. Both it and the duplicates were found by searching this site for [uniform jacobian](https://stats.stackexchange.com/search?q=uniform+jacobian). – whuber Nov 24 '21 at 14:33

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Your problem is that your integral is under the assumption that Y goes through 0 to $XY/X$ for a given X, which is not true because then Y would be allowed to be greater than 1 and in fact be unbounded. That's the reason why your integral blows up. You need to take care of this situation if you must go the Jacobian route.

Otherwise, as one comment states it's the easiest to get the answer with constructing CDF of XY as a double integral, and you'd do it through breaking the outer integral into two parts.

Aksakal
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