# PDF of function of 3 continuous, uniform random variables?

by Phillips101
Tags: continuous, function, random, uniform, variables
 P: 33 Hi. The question is: Given X, Y and Z are all continuous, independant random variables uniformly distributed on (0,1), prove that (XY)^Z is also uniformly distributed on (0,1). I worked out the pdf of XY=W. I think it's -ln(w). I have no idea at all how to show that W^Z is U(0,1). What do I integrate, how do I know how to combine the pdfs, how do I know what the limits are, what substitutions should I make if I need to make one? Etc, really. I just don't know how to tackle this sort of problem at all. The pdf I have for W came from a picture, not any real understanding of what I was doing. Thanks for any help :)
 P: 5 Fix x in (0,1). We could start with P(WZ ≤ x) = E[P(WZ ≤ x | W)]. Since Z and W are independent, we can calculate P(WZ ≤ x | W) by treating W as a constant. In this case, if W > x, then the probability is 0. Otherwise, WZ ≤ x iff Z ≥ ln(x)/ln(W), which has probability 1 - ln(x)/ln(W). Hence, \begin{align*} E[P(W^Z \le x \mid W)] &= E\left[{\left({1 - \frac{\ln(x)}{\ln(W)}}\right)1_{\{W\le x\}}}\right]\\ &= \int_0^x \left({1 - \frac{\ln(x)}{\ln(w)}}\right)(-\ln(w))\,dw. \end{align*} Now do the integral and check that the result is x.
P: 5
 Quote by Phillips101 The pdf I have for W came from a picture, not any real understanding of what I was doing.
We can do this the same way. If $$w\in(0,1)$$, then

\begin{align*} P(W\le w) &= P(XY \le w)\\ &= E[P(XY\le w \mid Y)]\\ &= E\left[{P\left({X\le\frac wY\mid Y}\right)}\right]. \end{align*}

If $$Y\le w$$, then the probability is 1; otherwise, it is w/Y. Thus,

\begin{align*} P(W\le w) &= E\left[{1_{\{Y\le w\}} + \frac wY1_{\{Y > w\}}}\right]\\ &= P(Y \le w) + \int_w^1 \frac wy\,dy\\ &= w - w\ln(w). \end{align*}

To get the density, we differentiate, which gives $$-\ln(w)$$.

P: 33

## PDF of function of 3 continuous, uniform random variables?

Thanks a lot, that's really very useful.

James

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