Simulation problem - inverse transform method not succeding

Question

Suppose 1 and 2 are i.i.d. (0,1). What's the distribution of − (1/3) ln(1) − (1/3) ln(1 − 2)?

I tried to solve this problem using inverse transform method but it did not lead me an answer. Can any one find a solution for this?

Answer 1

If $U_2$ is uniformly distributed on $[0,1]$ and independent of $U_1$, then so is $1-U_2$. Thus you are adding two independent random variables of the form $-\frac13\ln U$ with $U$ uniformly distributed on $[0,1]$. These have exponential distribution with parameter $3$. The sum of two exponential variables with the same rate parameter has Erlang distribution with shape parameter $2$ and rate parameter given by the parameter of the exponential variables (in this case $3$).