What is the distribution of $-\frac12\ln[(1-U_1)U_2]$ where $U_1,U_2$ are i.i.d uniform?

Question

How would you do the following? If you had two variables $U_1$ and $U_2$ are iid Unif$(0,1)$, what would the distribution of $-\frac12\ln[(1-U_1)U_2]$?

I thought it was $-1/\lambda$ which is Erlang $2(2)$, but am being told it is Erlang $2(-0.5)$.

Answer 1

Consider that if $Z=-\frac12\ln[(1-U_1)U_2]$, manipulating the expression of $Z$:

$$F_Z(t)=\mathbb{P}(Z \leq t)=\mathbb{P}(-\frac12\ln[(1-U_1)U_2] \leq t)=\mathbb{P}(U_2-U_1U_2 \geq e^{2t})$$

and the last one is the area in $\mathbb{R}^2$ of:

$$A_t=\{(x,y)\in ]0,1[ \times ]0,1[ : \frac{k}{1-x} \leq y \mbox{ where } k=e^{2t} \} $$

which can be easly computed using integrals.

Then to get the distrubution, just derivate $F_Z(t)$.