If I receive $20$ emails each hour, will it mean that the probability of receiving $10$ emails in $2$ hours will be $\frac{e^{-40} 40^{10}}{10!}$?

Question

If I receive $20$ emails each hour, will it mean that the probability of receiving $10$ emails in $2$ hours will be $\frac{e^{-40} 40^{10}}{10!}$?

I just learnt the Poisson distribution, so I wanted to confirm.

Answer 1

Let $X(t)$ be the random number of emails received per $t$ hours. If the average rate of emails is $\lambda = 20$ per hour, then $$X(t) \sim \operatorname{Poisson}(20t),$$ and $$\Pr[X(t) = x] = e^{-20t} \frac{(20t)^x}{x!}, \quad x \in \{0, 1, 2, \ldots \}.$$ So we observe your inbox for $t = 2$ hours, the probability you get $x = 10$ emails in that time span is $$\Pr[X(2) = 10] = e^{-40} \frac{40^{10}}{10!},$$ as you correctly wrote.