Joint distribution of 3 iid uniform random variables

Question

$X,Y,Z \sim U[0,1]$ and iid
How do I find the joint pdf of $(XY,Z^2)$

Answer 1

Let $X,Y,Z$ be i.i.d and uniformly distributed on the unit interval. Put $U = XY$ and $V = Z^2$ and $W=Y$. Hence $X = U/W$ and $Y=W$ and $Z=\sqrt V$, so that $$f_{U,V,W}(u,v,w) = \frac{1}{2w\sqrt v}$$ for $0\leq v,w\leq 1$ and $0\leq u \leq w$. Hence $$f_{U,V}(u,v) = \int_u^1 \frac{1}{2w\sqrt v}\,dw = \frac{1}{2\sqrt v}(e-\log u)$$