Expected value in a poisson distribution when there's a maximum possible value Bit confused with a certain part of this q:
A shop has four copies of the magazine B delivered each week. The demand follows a Poisson distribution with mean 3.2 requests per week.
Find the expected number of books sold each week.
So the answer is 2.8, but I'm not sure what you're supposed to do with a Poisson distribution that has a maximum possible value. I tried to work out the mean using the poisson E(x) formula but going up to 4 not infinity, but then I got 1.928. When I did the sum up to 5 I got 2.5 which was a bit closer to the answer but I was just guessing, I'm just not sure what the method is or how you're supposed to approach it.
 A: Your $\sum\limits_{x=0}^4 \dfrac{xe^{-3.2}3.2^x}{x!} \approx 1.928$ deals with what happens when demand is $4$ or fewer
but you need to add to this what happens when demand is $5$ or more.  In that situation $4$ are sold so you need to add $\sum\limits_{x=5}^\infty \dfrac{4e^{-3.2}3.2^x}{x!}$. That infinite sum is a little annoying, but  is $= 4-\sum\limits_{x=0}^4 \dfrac{4e^{-3.2}3.2^x}{x!} \approx 0.878$ 
Adding these two together gives about $2.806$
A: I believe that there is an unspoken assumption that you're supposed to find mean number of books sold at equilibrium. This is essentially a a queue with arrival process with constant rate $4$ per day and service process described by this Poisson probabilities.
You can imagine states of the shops storage as non-negative integers and transitions between them form a Markov chain. If we order them from left to right from zero to infinity we get that transitions to the right occur in steps of 4 this should give you a recurrence relation for equilibrium distribution of the number of books in the store, from that you can compute the number of books sold.
This sound like quite a laborious approach.
Alternatively this can be described as a G/M/1 queue which sells books continuously at rate 3.2/week with exponential waiting times and books arrive at constant intervals. It seems that there should be some clever trick in queueing theory to finish this off from there...
