# Probability of event on Poisson process arrival

I have the following problem:

A mail server, sends emails with rate $7.5/\text{hour}$.

What is the probability that it will send exactly $5$ emails in one hour?

To solve it, I did the following using a Poisson distribution:

$$\frac{7.5^5 e^{-7.5}}{5!}$$

or with sage math:

def dpoi(x):
return (7.5^x * (e^-7.5)) / factorial(x)

dpoi(5)
0.109374594682555


is that right?

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Yes, that's correct. –  Eckhard Dec 30 '12 at 15:39
Hey @Eckhard. You should read it meta.math.stackexchange.com/questions/6883/… you can add your comment as answer and I do accept it. –  VP. Dec 30 '12 at 15:51

## 2 Answers

As suggested I repeat my comment as an answer:

Yes, that's correct.

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This is a mere generalization of Eckhard's answer.

If $X \sim Poisson(\lambda)$, then the pdf of $X$ is $P(X=k)=\frac{e^{-\lambda} \lambda^{k}}{k!}$, and this is the same as asking 'what is the probability that exactly $k$ events happen in a given time interval'

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thank you for improving the knowledge base :-) –  VP. Dec 31 '12 at 9:32