# how to prove the probability distribution of a Poisson random distribution over a time?

My question may seem so simple but I have problem to prove it. Suppose that $$X$$ follows a Poisson distribution with parameter $$\lambda$$ which is defined per each period of time. How we can prove that the probability distribution of arrival over $$t$$ follows a Poisson with parameter $$\lambda.t$$? Thanks

• You may have to make a further assumption that they are generated by a homogeneous Poisson process. Then the result is essentially immediate. – Henry Nov 5 at 20:06
• It seems trivial but I like to know the proof. thanks – Katatonia Nov 5 at 21:28
• Wikipedia has an explanation, though perhaps closer to a definition than a proof. – Henry Nov 5 at 21:39
• My post at stats.stackexchange.com/a/215253/919 derives this result from basic principles and definitions, without any advanced math. – whuber Nov 5 at 22:03

The characteristic function of a Poisson distribution with rate $$\lambda$$ is $$\varphi(s)= \exp(\lambda(e^{is}-1))$$
So the sum of $$t$$ i.i.d. such random variables is $$\exp(\lambda(e^{is}-1))^t = \exp(\lambda t(e^{is}-1))$$ which is the characteristic function of a Poisson random variable with parameter $$\lambda t$$. That shows your desired result for positive integer $$t$$
Now consider rational $$t=\frac m n$$. We now know that the characteristic function of a Poisson random variable with parameter $$\lambda m$$ is $$\exp(\lambda m(e^{is}-1))$$ and this is equivalent to the characteristic function of the sum of $$n$$ i.i.d. random variables each with characteristic function $$\sqrt[n]{\exp(\lambda m(e^{is}-1))}=\exp(\lambda \frac m n(e^{is}-1))$$, i.e. $$\exp(\lambda t(e^{is}-1))$$ which is the characteristic function of a Poisson random variable with parameter $$\lambda t$$. That shows your desired result for positive rational $$t$$
You can extend from the positive rationals to positive real $$t$$ by continuity and thus get your final result. You could use moment generating functions or probability generating functions instead of characteristic functions and still have essentially the same demonstration.