Let's suppose there are two buses number $86$ and $98$. They draw up at the bus stop under the Poisson distribution with intensities $3$ and $5$ times per hour. (a) What's the expected length of time after the $15$th bus will arrive?, (b)What's the probability that exactly $12$ buses will arrive within $3$ hours?
Poisson distribution $P(N(t)=j)=\frac{(\lambda t)^j}{j!}e^{-\lambda t}$. We have that $j =3$ or $j =5$. Do I just substitute $j =3$ and $\lambda t=15$ and we immediately have (a)? I'm aware that's a really easy exercise but I somehow don't really know how to approach this one. I'm also not sure how to approach subpoint (b). I'll be thankful for any tips and help.