Given two iid random variables $X,Y$ where $X \sim exp(\lambda)$ I have calculated that the pdf of $Z:=X+Y$ is given by $f_{X+Y}(z)=\lambda^2ze^{-\lambda z}$ for $z \geq 0$.
Now I want to calculate the pdf of $(X,Z)$. I know that if $X$ and $Z$ were independent I would use $f_{X,Z}(x,z)=f_X(x)f_Z(z)=f_X(x)f_{X+Y}(z)$.
How do I check independency and if they are not independent, how do I calculate the joint pdf of $X$ and $Z$? Some help would be much appreciated.
They are not independent. Intuitively the fact that $Z \geq X$ tells you that $X$ and $Z$ are dependent. Let $0 \leq x \leq z$. $P\{X\leq x, Z\leq z\}=P\{X\leq x, Y\leq z-X\}$. You can compute this by conditioning on $X$. You get $EI_{\{X\leq x\}} [1- e^{-\lambda (z-X)}]$ which is $\int_0^{x} [1-e^{-\lambda (z-t)}] \lambda e^{-\lambda t} \, dt$. Can you compute this integral and take derivative to get the density function?