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I have small 2 questions:

  1. Is the sum of two Poisson variables only then a Poisson variable, if they are independet?

  2. If $X,Y$ are Poisson. And $X=Y$. Is $Z= X-Y$ also Poisson (if they are independet or not)?

WaldoRozir
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  • Either you have a typo in 2 or $Z=0$. Also you should give some context like your thoughts on the problem. Nonetheless I'll give a small hint for 1: usually if you're thinking of whether you can remove an independence assumption entirely a good first test is for the variables $X$ and $X$, which are a good example of two non-independent variables with the same distribution. – spaceisdarkgreen Jul 06 '17 at 01:55
  • @spaceisdarkgreen For 2) if $X,Y$ are two Poisson variables with parameter $\delta$ a $\gamma$. If $X=Y$ is it then true that the difference $Z=X-Y$ is not Poisson? (please look at both case: Independent and non-Independent) – WaldoRozir Jul 06 '17 at 02:15
  • Huh? If $X=Y$ then $Z = X-Y = 0.$ Not sure how to parse the part about the parameters but if they have different distributions, how can $X=Y$? – spaceisdarkgreen Jul 06 '17 at 02:16
  • @spaceisdarkgreen And is this not the same as a Poisson variable which is distributed with parameter $0$? Edit: no different distribution. I only want to know the cases when $X,Y$ are independet or not. – WaldoRozir Jul 06 '17 at 02:18
  • I suppose it is (usually $\lambda =0$ is not allowed, but I see no reason not to admit it as a degenerate case). Still this is a strange question, and not well-founded if the parameter for $X$ and $Y$ is different. – spaceisdarkgreen Jul 06 '17 at 02:21
  • @spaceisdarkgreen Please let me reformulate my question: If $X,Y$ are Poisson distributed, is $X-Y$ also Poisson distributed or not? (including the case that the difference could be Poisson with $\lambda = 0$)? – WaldoRozir Jul 06 '17 at 02:24
  • In that one uninteresting case where $X=Y$, yes. Otherwise... think about it. If $X$ and $Y$ are independent, both with parameter $\lambda,$ what does the distribution of $X-Y$ look like? Don't calculate it.. just what does it look like? – spaceisdarkgreen Jul 06 '17 at 02:33
  • To first question, see example here: https://math.stackexchange.com/a/2168580 – NCh Jul 06 '17 at 03:16

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Regarding the 2nd question, assuming you didn't mean $X = Y$, you can find the distribution of $X - Y$ here. It is called a Skellam distribution and looks quite complicated.