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