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If X and Y are uniform(0,1) then what is the distribution of $X^kY^m$ for some integers k and m?

Shakil
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1 Answers1

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$$ p(X^k Y^m=t)=\int_0^1dx\int_0^1 dy\ \delta(t-x^ky^m)\ , $$ and using the Dirac delta to kill the $x$-integral$$ p(X^k Y^m=t)=\int_0^1 dy\ \Theta\left(0\leq (t/y^m)^{1/k}\leq 1\right) \frac{1}{y^m k ( (t/y^m)^{1/k})^{k-1}}\ , $$ $$ =\Theta(0\leq t\leq 1)\int_{(1/t)^{-1/m}}^1 \frac{dy}{y^m k ( (t/y^m)^{1/k})^{k-1}}=\boxed{\Theta(0\leq t\leq 1)\frac{t^{1/k}-t^{1/m}}{(k-m)t}}\ . $$ Note that the result is symmetric upon the exchange $m\to k$ as it should. It is also correctly normalized, $\int_0^1 dt\ p(X^k Y^m=t)=1$ and reduces to $-\log(t)$ for $k=m=1$ as it should (check this answer product distribution of two uniform distribution, what about 3 or more).

Community
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Pierpaolo Vivo
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  • This misses the hypothesis that $(X,Y)$ should be independent and relies on unconventional notations since $P(X^kY^m=t)=0$ for every $t$. To sum up, do not give this to a maths'TA as solution to some work they asked. – Did Sep 18 '16 at 15:39