Law of total probablity

Question

If I recieve an email as Poisson distribution with parameter k, poi(k) during a time with geometric distribution with parameter p, geo(p). Then the total emails depends on the time t. How do I find that distribution? Law of total probability?

Answer 1

How do I find that distribution? Law of total probability?

Yes. Do that.

Let $N$ be the total count of emails you receive, and $T$ the count of time periods available.   So you are given:

$${N\mid T\sim\mathcal{Pois}(kT)\\T\sim\mathcal{Geo}_1(p)}\implies{\mathsf P(N=n\mid T=t)=\dfrac{(kt)^n\mathrm e^{-kt}}{n!}\,\mathbf 1_{n\in\Bbb N}\,\mathbf 1_{t\in\Bbb N^+}\\\mathsf P(T=t)=(1-p)^{t-1}p\,\mathbf 1_{t\in\Bbb N^+}}$$