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Okay, so I have this problem that states $X$~$Exponential(2)$, and $Z$~$Exponential(2)$ where Z is independent from $X$. The problem that I'm trying to solve wants me to find the distribution $X+Z$ and find the $Var(X-Z)$.

The first part of the question is to find the distribution of $X+Z$. I'm not sure if I need to calculate something from this or just realize that when we are adding to Exponential distribution, we can use the Gamma distribution. So to the first question my answer would be: $X+Z$~$\Gamma(2,2)$.

The second question is to find the variance of $X-Z$. Here my first thought is to use the definiton of variance for Gamma distributions, which is $Var(X)=\alpha/\lambda^2$. In this case my answer would be $Var(X-Z)=2/2^2$$=1/2$.

I would, of course, like to know if I'm right or wrong. And if I'm wrong what would you recommend that I do instead? Btw I'm pretty new to statistics and trying to learn.

Leonardo
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    If you know how to use moment generating functions, this is trivial. Now, you've appended a second problem. $Var(X - Z) = Var(X)+Var(Z)$ because $X$ and $Z$ are independent. – BruceET May 20 '18 at 00:33
  • I have read a bit about it, but I'm not that familiar with it. – Leonardo May 20 '18 at 00:36
  • Now might be a good time to learn about MGFs. // Simulation in R of a million realizations of the RV $X-Z:$ m = 10^6; x = rexp(m,2); z = rexp(m,2); var(x-z) returns 0.5009375, which should be accurate to at least two places. It illustrates the formula in my previous comment. Until you're sure about the gamma part, it seem best to get $Var(X - Z)$ from what you know about exponentials. – BruceET May 20 '18 at 00:44
  • Okay, thaks a lot. It makes more sense for me to find the variance through my knowledge from exponentials. But about the first question, I'm still in doubt if I just need to realize that $X+Z$~$\Gamma(2,2)$ or to calculate something? – Leonardo May 20 '18 at 01:09
  • You have to prove that the sum of iid exponentials is gamma. If you just use this fact, then there's nothing left in the problem to show. And how do you 'realize' it is gamma without 'calculating ' it? – StubbornAtom May 20 '18 at 03:14
  • Roughly: MGF of $X \sim \mathsf{Exp}(rate=\lambda)$ is $M_X(t) = \frac{\lambda}{\lambda-t}.$ MGF of $Y\sim\mathsf{Gamma}(shape=\alpha,rate=\lambda)$ is $M_Y(t) = (\frac{\lambda}{\lambda-t})^\alpha.$ MGF of $Z \sim f_Z$ is $M_Z(t) = E(e^{tZ}).$ If $X,Y$ indep, then $M_{X+Y}(t) = M_X(t)M_Y(t).$ Check your text or Wikipedia on 'moment generating function' for details. MGFs of 'exponential' and 'gamma' are given at Wikipedia pgs for these distn's. Sooner or later you need to know about MGFs. "Resistance is futile." – BruceET May 20 '18 at 07:03
  • Also, see this related page. – BruceET May 20 '18 at 07:35
  • I have read it somewhere in a textbook that Gamma, Exponentials and Normal distribution are related. But that you I'll look more into that – Leonardo May 20 '18 at 15:58
  • Thank you BruceET, it helped a lot – Leonardo May 20 '18 at 15:58

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