Customers arrive randomly and independently at a service window, and the time between arrivals has an exponential distribution with a mean of 12 minutes. Let X equal the number of arrivals per hour. What is P[X = 10]?
Now the solution to this problem uses this logic:
If the time between arrivals is exponential with mean 12 minutes and arrival times are independent then the number of arrivals in any single minute is Poisson with mean 1/12. Since the sum of independent Poisson variables is also Poisson, the number of arrivals in an hour will be Poisson with mean: 5
My question is how can an exponential distribution be related to a Poisson PMF? Thank you