I'm trying to solve the next excercise.
Let $X_1,...,X_n$ iid random variables with pdf $$f(x,\theta)=\frac{\theta}{(x+1)^{\theta+1}}, x>0$$ with a parameter $\theta>0.$
Find the distribution of the statistic $T$ given by $$T=\sum_{i=1}^{n}\ln(1+X_i)$$
In my attempt I got stuck:
$\begin{align}P[T<x]&=P\left[\sum_{i=1}^{n}\ln(1+X_i)< x\right]\\ &=P\left[e^{\sum_{i=1}^{n}\ln(1+X_i)}< e^x\right]\\ &=P\left[\prod_{i=1}^n(1+X_i)< e^x\right] \end{align}$
Any help is appreciated. Thank you.
