Let $\displaystyle X_1,X_2$ be two i.i.d. $\displaystyle Exp(\lambda)$ random variables and $\displaystyle R$ is a $\displaystyle Ber(\frac{1}{2})$ variable. How to show that the distribution functions of $\displaystyle X_1-X_2$ and $\displaystyle RX_1-(1-R)X_2$ are identical?