I'm trying to learn how to use Poisson distribution for questions involving changing intervals. I have the following exercise:
Ron gets on average 5 points a day (24 hours) in his computer game. What is the probability for getting zero points during a) one day b) one afternoon (8 hours)?
So for a I have the following formula: $\frac{5^0e^{-5}}{0!}$
However when I try to adjust it to include only one third (8/24 hours), I encounter the situation where I should take a factorial of $\frac{5}{3}$, which is not integer. Do you have any tips how to proceed?
Invoking the reproducibility property of poisson distribution, the number of point get in 8 hours follows a $Po\left(\frac{5}{3}\right)$
Thus
$$\mathbb{P}[X=0]=\frac{e^{-5/3}\left(\frac{5}{3}\right)^0}{0!}=e^{-5/3}$$