A question about conditional probability and independent random variables

Question

I need some help with this following question: Given two independent random variables $X\sim \exp(1),Y\sim \exp(1)$ $Z \sim X+Y$. We want to calculate $f_{X|Z}(x,z)$.

So first I'd like to calculate $F_{X,Z}$

$$F_{X,Z}(x,z) = P(X\le x, Z\le z) = P(X\le x, X+Y\le z) = P(X\le x, Y\le z-X).$$

My question is, is it okay to say that: $$P(X\le x, Y\le z-X) = P(X\le x)\cdot P(Y\le z-X),$$ because they are independent?

Answer 1

Hint

Let $x\geq 0$. Using Total probability formula and the fact that $X$ and $Y$ are independents yields

\begin{align*} \mathbb P\{X\leq x,Y+X\leq z\}&=\int_{\mathbb R}\boldsymbol 1_{\{s\leq x\}}\mathbb P\{Y\leq z-s\}f_X(s)\,\mathrm d s\\ &=\int_{0}^x \mathbb P\{Y\leq z-s\}f_X(s)\,\mathrm d s. \end{align*}