# PMF of two Poisson RV subtracted from each other [duplicate]

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.

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

• 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? – gung Sep 20 '13 at 20:53