What is the probability distribution of a random variable with distribution function $F_{Y-X}(z)=1-\frac{\nu}{\nu+\lambda} e^{-\lambda z}$?

Question

I have this:

$F_{Y-X}(z)=1-\frac{\nu}{\nu+\lambda} e^{-\lambda z}$.

I have to find a probability distribution to work with and then obtain $E(Y-X)$ and some other data based on the distribution function given above. I've tried by just founding the first derivative but it turned out not to be a true density function.

I can notice the distribution function is quite similar to that of an exponential distribution, but I can't figure out any way I can get a density function out of it.

Any ideas?