Question:
A server recieves a document every 4 requests each hour.
S ~ Poisson(4)
For simplicity we'll say the server only runs from 5pm to 8pm per day.
One day, you know that during the hours of 6pm and 8pm, 10 requests will come.
Given that the 10 requests will come, what is the expected value between 5pm and 8pm?
Where I am at so far:
I believe that the server, from 5pm to 6pm will have a Poisson distribution of:
Since there's no reason to believe it'll be different, but I'm unsure what to do with the 6pm-8pm range. Naively this might work:
But with one observation, I don't think you can make that assertion that the lambda for that period is 10.
The combined distribution should be something like:
And then the expected value will be just that 4 + ? parameter
It is known with certainty that $10$ that occur between 6 and 8pm, so the number that arrived in those hours is not random. By the properties of a Poisson process, this knowledge has no effect on the distribution of number of arrivals between 5 and 6pm, so that is Poisson-distributed with mean $4.$ So the distribution of total arrivals is 10 + Poisson(4), i.e. a constant added to a poisson variable. This has mean $10+4=14.$