<s>[INST] Find the distribution of random variable $XY+X+Y+1$ X and Y are iid with density $f(x)=\frac{1}{(1+x)^2}I_{(0,\infty)}$.
Find $P(Z\le z)$ where $Z=XY+X+Y+1$
my effort:
$P(Z\le z)=P((x+1)(y+1)\le z)=P(x\le \frac{z}{y+1}-1)=\int_{0}^{\infty}\int_{0}^{\frac{z}{y+1}-1}\frac{1}{(1+x)^2}dxdy=\int_{0}^{\infty}1-\frac{y+1}{z}dy=...?$
If I integrate it once more the result is $\infty$, which is obviously wrong - where have I done a mistake?
 [/INST] Note that $y = \dfrac{z}{x+1}-1 \le z-1$ for any value of $x \ge 0$. 
So, the bounds for the integrals should be $0 \le y \le z-1$ and $0 \le x \le \dfrac{z}{y+1}-1$. 
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