I have two variables $X_1$ and $X_2$ are IIDs distributed with the common distribution $X \sim Exp(\lambda)$ for some $\lambda >0$. I was working to find a distribution of $|X_1−X_2|$.
I used convolution to find the function $f_Z(z)$ as \begin{align} f_Z(z) &= \int_{-\infty}^{\infty}f_x(x)*f_y(x-z)dx , \ for \ X_1 > X_2 \ \& \\ f_Z(z) &= \int_{-\infty}^{\infty}f_{-x}(x)*f_y(z+x)dx , \ for \ X_1 < X_2 \end{align}
Is this the correct method to do it & how to find the range of $z$?
By symmetry between $X_1$ and $X_2$ we can write $$P(|X_1-X_2| \leq z)$$ $$=2P(X_2 \leq X_1\leq X_2+z)$$ $$=2\int_0^{\infty}\int_{x_2}^{x_2+z} \lambda^{2}e^{-\lambda x_1}e^{-\lambda x_2}dx_1dx_2.$$ for $z>0$. I will let you do the computation.