n this problem we show directly that the sum of independent Poisson random variables is Poisson. Let J and K be independent Poisson random variables with expected values α and β respectively, and show that N = J + K is a Poisson random variable with expected value α + β.
![](https://test.transtutors.com/qimg/d606009c-62f7-4b0a-b070-b47cf8e80475.png)
And then simplify the summation by extracting the sum of a binomial PMF over all possible values.