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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?

Juan Esaul González Rangel
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  • Are $X$ and $Y$ known to be exponential RVs with parameters $\nu$ and $\lambda$ (or vice-versa)? If so you can find the density of $Z=Y-X$ through the transformation technique. Alternatively, you may differentiate $F_Z(z)$ given here and find the region $A\subset \mathbb{R}$ such that $\int_A f_Z(z) dz =1$, which may be a little trickier than the former idea. – Nap D. Lover Nov 09 '19 at 19:35
  • Actually, having the distributions of $Y$ and $X$ individually would make computing $\mathbb{E}(Y-Z)=\mathbb{E}(Y)-\mathbb{E}(X)$ trivial. Where does this problem come from anyway, and is this the full set up or is there any information you did not include here? – Nap D. Lover Nov 09 '19 at 20:14
  • Yes, thank you, I've just realized I was doing it the dumb way, but the problem was proposed liked that. I have the source in spanish, give a while and I will translate, I'm finding it very tough. – Juan Esaul González Rangel Nov 09 '19 at 20:18
  • Here is the whole problem set link We were given the result of (1.1.1) and were supposed to work using it. – Juan Esaul González Rangel Nov 09 '19 at 20:42
  • I am not so good with English, so please let me know if there's something you don't understand. – Juan Esaul González Rangel Nov 09 '19 at 20:43
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    You can find the distribution of $Y-X$ here: https://math.stackexchange.com/questions/115022/pdf-of-the-difference-of-two-exponentially-distributed-random-variables – Math1000 Nov 10 '19 at 00:47
  • Thank you @Math1000, it was very helpful! – Juan Esaul González Rangel Nov 10 '19 at 20:08

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