arrival time poisson processes The arrival times of customers in a bakery can be modeled by a Poisson process (Nt)t≥0 with some rate λ > 0. On average, there are four arrivals per unit of time.
I want to ask about the question above what does it mean by "four arrivals per unit of time" and how can I get the rate(lamda) by just knowing the above information?
Thank you!
 A: Your question is a good one. It is somewhat vague to say that the number of customers in a bakery can be modeled by a Poisson Process. To understand what that means, imagine this exercise:


*

*Count the number of people that will arrive at the bakery in the next 5 minutes. Say you saw 3 customers in the first 5 minutes. Record that number. Now repeat the process. Over the next 5 minutes count how many customers arrive at the bakery. This time only 2 customers arrived. Over the following 5 minutes - 4 customers, in the 5 minutes after that - 2 customers, etc. Repeat this process for a total of say 1000 times. 

*Group the 1000 observations by their values. You may have ended up with 100 of the 1000 5-minute intervals with a value of 0 (no customers arrived during those 5 minute intervals), 200 5-minute intervals with a value of 1, 300 5-minute intervals with a value of 2, 350 5-minute intervals with a value of 3, and 50 5-minute intervals with a value of 4 (4 customers arrived during each of these 5-minute intervals).

*Call X a random variable which will take as its value the number of people arriving in a 5-minute interval. In your case, the random variable takes only values 0-4. The fact that the problem says that the number of customers in this bakery can be modeled by a Poisson process means that if you calculated the relative frequency for each value of X, it should be relatively close to that obtained by the theoretical Poisson distribution function:



where lamda stands for the expected number of customers during any 5-minute interval (as Ed V pointed out); this is what you are given as the average number of arrivals per unit of time.
Let's look at a couple values and compare the relative frequency from our sample with the predicted probability by the Poisson distribution function.
X=0
In our sample, we saw observed 0 customers in 100 of 1000 5-minute intervals, hence with a relative frequency of 0.10. On the other hand, the Poisson distribution predicts that

we would see 0 customers during 1.83% of the 5-minute intervals.
X=1
In our sample, we saw observed 1 customer in 200 of 1000 5-minute intervals, hence with a relative frequency of 0.20. On the other hand, the Poisson distribution predicts that

we would see 1 customer during 7.32% of the 5-minute intervals.
Here is a picture from Wikipedia of what a theoretical Poisson distribution with a lamda=4 would look like.

https://en.wikipedia.org/wiki/Poisson_distribution
In my contrived example, it doesn't look like the number of customers at the bakery is modeled well by the Poisson. What could be wrong (aside from the fact that these numbers were completely made up)?
The fact that the arrival of customers at a bakery can be modeled by a Poisson process says that there is some interval of time, (like the 5-minute long interval we used above), such that if we counted the number of customers arriving during those intervals, and calculated those relative frequencies, they would be close to those predicted by the Poisson distribution function. But notice that we are not told what that interval is. It could be 2.5 minutes, 0.7 minutes, or 0.05 seconds. This magic interval length has to be such that if we imagined dividing it into n non-overlapping sub-intervals, where n is large: 
a) the probability that 2 or more events (in our case the arrival of customers) occur in any given sub-interval is essentially 0, 
b) the events occur independently, and 
c) the probability that an event occurs during a given sub-interval is constant over the entire interval (dependence between intervals ok but not within intervals).
The Poisson Process most likely models well the arrival of customers in this bakery for only 1 magical interval length. For all other interval lengths, the Poisson Process most likely does not model it well, as some of the assumptions above (such as independence) would likely not hold. So to say that the Poisson process models something well without stating the interval length is to leave out a very interesting, crucial piece of information, that should prompt the student to wonder what interval might work in the problem at hand.
