# Is there a probabilistic (not analytical) argument for why the sum of independent Poissons is Poisson?

I was just wondering if there was a simple probabilistic argument for why a Poisson random variable with parameter $\lambda_1$ plus a Poisson random variable with parameter $\lambda_2$ (both independent) is a Poisson random variable with the sum of the parameters as a parameter.

I know one exists for binomial but can't think of one for Poisson.

• If $\lambda_1\neq \lambda_2$, they're not iid, and if $\lambda_1=\lambda_2$, why distinguish them? But you can argue directly from the Poisson process itself (without needing the two Poissons to be identically distributed) – Glen_b -Reinstate Monica Nov 4 '15 at 1:25
• Really? What about the poisson process allows for the sums to be poisson processes? – jchaykow Nov 4 '15 at 2:10
• Because the superposition of the two processes (which count of events is the sum of the two Poisson counts) satisfies all the original conditions for the Poisson process. – Glen_b -Reinstate Monica Nov 4 '15 at 2:14

As Glen_b noted in comments, think of a Poisson variable as counting the number of events for a Poisson process of intensity $\alpha$ in a fixed window of size $w$: the Poisson parameter for this variable is $\alpha w$.
Suppose now there are two independent Poisson processes separately running in the same window, one of intensity $\alpha_1$ with $\alpha_1 w = \lambda_1$ and the other with intensity $\alpha_2$ with $\alpha_2 w = \lambda_2.$ The sum of these counts clearly is a Poisson process of intensity $\alpha_1 + \alpha_2$ because it trivially satisfies all the requirements of Poisson process (the count is proportional to the size of the window, events in non-overlapping windows are independent, and the chance of a count exceeding $1$ grows vanishingly small as the window size shrinks to zero). Therefore the sum of counts is a Poisson random variable with parameter $(\alpha_1+\alpha_2)w = \lambda_1+\lambda_2,$ QED.
If you're satisfied with the fact that binomials sum to binomials when their parameter $p$ is the same, then you can view Poisson distributions as binomials with rare success that scales like $p\approx c/n$. This correspondence is exact in the limit so two Poissons sum to another Poisson immediately follows from the binomial case. In the case where the binomials have different probabilities $p_1,p_2$, they sum to a binomial with an adjusted parameter, whose adjustment approaches $p_1+p_2$ as both $p$'s get very small.