Random variables X, Y, Z are independent with uniform distribution from [0, 1]. find density of $XY^{Z}$
actually I'm stuck because power is a function (Random variable). I understand how to find density of XY using substitution, calculating Jacobian and integration. but after I get density of XY I have no idea how to find $XY^{Z}$, even though I know how to find density of composition $\phi(\xi)$ I got no idea how to get inverse function here... please help me out
Note that if $X \sim \operatorname{Uniform}(0,1)$, then $$-\log X \sim \operatorname{Exponential}(1).$$ The proof of this is left as an exercise.
Next, consider $$\log (XY^Z) = \log X + Z \log Y.$$ What is the distribution of $Z \log Y$ when $Y, Z \sim \operatorname{Uniform}(0,1)$ and are independent? Edit your question to include your work in order to proceed further.