I would like to model the amount of phone calls at each time of the day. The phone calls should follow a poisson distribution and at 12:00 there should be the peak.
So, semantically what I would like to have is a mapping like this:
00:00:00 --> 5 calls
00:00:01 --> 7 calls
12:00:00 --> 723 calls
16:00:00 --> 55 calls
23:59:59 --> 5 calls
My approach was to try and group the calls in 10sec buckets so 24h hours is divided into 14400 buckets. Then I created a poisson PDF which peaks in the middle of the series, therefore I used a $\lambda = 7200$. My thought was that I could then just multiply the amount of calls (e.g. 100000) I would like to have distributed with the according probability of my poisson PDF.
PDF(1)*100000
PDF(2)*100000
...
PDF(14400)*100000
But with this approach the calls get distributed only during a very small range in the middle of the series instead over the whole series. Can somebody see where the mistake in my approach lies?
edit: I think figured out where my mistake is. I need to generate the amount of calls for each bucket with a distribution other than a poisson (e.g. something more like a normal distribution) because poisson is not suitable to model this distribution. And then in the next step a poisson PDF for each bucket has to be created and used.