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Transformation of the exponential distribution

At least I think that's the sort of thing the question's aiming at...

*John and Jane have arranged to meet at a given time, and each, independently, will be late by an amount of that time that is exponentially distributed with parameter $\displaystyle \lambda$. Let $\displaystyle X$ and $\displaystyle Y$ denote the amount of time that John and Jane are late respectively.*

*Suppose that we are interested in the length of time that Jane arrives before John, that is, in $\displaystyle W=X-Y$. For $\displaystyle w\geq0$, show that*

*$\displaystyle F_W(w)=P(X-Y\leq w)=1-\frac{1}{2}e^{-\lambda w}$*

Seriously do feel like this isn't all that complicated, but I just keep hitting a wall.

I attached the past paper this question comes from - it's question B6, part (b), part (ii).