Cars arrive at a toll booth at a mean rate of 15 cars every half hour according to a poisson process. Find the probability that the toll collector will have to wait longer than 36.78 minutes before collecting the eleventh toll.
Please Help!!!
I will use the formulation of Poisson from wikipedia.
$\displaystyle \lambda$ is the expected number of occurrences over the particular interval of interest. So in our case the expected rate of occurrence is $\displaystyle 15/30=0.5$ cars per minute. So the expected number of cars in 36.78 minutes is: $\displaystyle \lambda=15/30*36.78=18.4$
If $\displaystyle N$ is the number of cars that arrive in 36.78 minutes, then the question can be rephrased to: what is the probability that $\displaystyle N<11$
The PDF is:
$\displaystyle f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!}$
So
$\displaystyle \mathrm{P}(N<11) = \sum_{k=0}^{10}\frac{\lambda^k e^{-\lambda}}{k!}=\ldots $