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Suppose $a$ and $b$ are independent Poisson RV with $\lambda_a$ and $\lambda_b$ respectively.

Find the pmf of $c = a - b$. Is $c$ a Poisson RV?

Characteristic function of $c(w)$ would be $$E[e^{iwc}]=E[e^{iw(a-b)}]=E[e^{iwa}]*E[e^{-iwb}]=e^{-(\lambda_a - \lambda_b)(1-e^{iw})}$$

this is a Poisson RV.

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marked as duplicate by whuber Sep 20 '13 at 20:06

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ The answer is not correct. Intuitively, the difference of two poisson distribution cannot be another poisson distribution since it can take negative values. The correct pmf is named a Skellam distribution with a rather complicated analytic expression. $\endgroup$ – Mr Renard Sep 20 '13 at 20:15
  • $\begingroup$ The answer at the linked thread discusses the possibility that $a$ & $b$ are independent Poissons, @user30523. If the answer there is insufficient, can you edit this to specify what it is you still don't understand? $\endgroup$ – gung Sep 20 '13 at 20:53