I have the following problem:
Let $\{ X_t : t \geq 0\}$ a Poisson process with parameter $\lambda$ and let $a > 0$ a constant. Show that $\{ X_{at} : t \geq 0 \} $ is a Poisson process with parameter $\lambda a $.
I don't really know how to begin and end the proof, it is an obvious result, what I would just do is define a parameter $\lambda ^* = a \lambda $ and just rewrite the pdf for a Poisson distribution. I mean, it doesn't really matter what $a$ it is, as long as it's $a<\infty $ then it could be the initial parameter $\lambda$ we're given. Though, I don't kow how to formalize this result, any help is really appreciated.