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

Question

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

Answer 1

This is an attempt at a detailed demonstration. Whether it is a proof or not is up to you.

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.