# Poisson Process-theory questions

Hey guys, what's up? I have some questions regarding the Poisson Process. I checked some threads, but not all, so forgive me if these questions have been answered before.

1)Lets say I am given some some tables or/and graphs or generally speaking some data and I am asked to find out if the Poisson model can describe it. What exactly should I look for?
Should I focus on these 2?
a)Should I prove the Poisson distribution
.
?
b)Should I prove that it has independent increments?

2)If i want to show that Poisson Process isn't suitable, should I just show that mean!=variance or is this not enough?

3) As far as the parameter λ is concerned. Ιf It changes over time, then the Poisson Process in non-homogeneous and if it is a constant then it is homogeneous, right?
So If I have some data(events) in a year, in order to be homogeneous, the following equation must hold----->
(x units in t1)/t1=(z units in t2)/t2 ? And if with this logic, I find for 2 periods(lets assume that these periods are overlapping,just to see if there is a problem with that :P ) of time, λ= 800000 units/month and
λ'=800001 units/month then what? Are these considered equal even though they aren't, and if yes, how to know what difference isn't bad enough?

## Answers and Replies

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Ray Vickson
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Dearly Missed
Hey guys, what's up? I have some questions regarding the Poisson Process. I checked some threads, but not all, so forgive me if these questions have been answered before.

1)Lets say I am given some some tables or/and graphs or generally speaking some data and I am asked to find out if the Poisson model can describe it. What exactly should I look for?
Should I focus on these 2?
a)Should I prove the Poisson distribution
.
?
b)Should I prove that it has independent increments?

2)If i want to show that Poisson Process isn't suitable, should I just show that mean!=variance or is this not enough?

3) As far as the parameter λ is concerned. Ιf It changes over time, then the Poisson Process in non-homogeneous and if it is a constant then it is homogeneous, right?
So If I have some data(events) in a year, in order to be homogeneous, the following equation must hold----->
(x units in t1)/t1=(z units in t2)/t2 ? And if with this logic, I find for 2 periods(lets assume that these periods are overlapping,just to see if there is a problem with that :P ) of time, λ= 800000 units/month and
λ'=800001 units/month then what? Are these considered equal even though they aren't, and if yes, how to know what difference isn't bad enough?

For (1): what you should (or can) do depends on what is the form of the data. If you have a (hopefully long) sequence of actual, individual arrival times, you can test the inter-arrival times for (i) independence; and (ii) exponential-distributeness (if that is a word). Just testing independence is not enough; a renewal process has independent increments but can be very far from a Poisson process. On the other hand, if all you have are, say, something like a frequency distribution of arrivals in time intervals, you cannot recover individual inter-arrival times so cannot do anything other than distribution-fitting (to see how well the histogram matches that of a Poisson distribution).

(2) Testing mean vs. variance is a good start, but of course is not conclusive. You can start with a genuine Poisson process, generate a random sample from it, and then compare the sample means and variances. They will never match exactly!

Anyway, if instead you have inter-arrival information, you should have $\text{mean} = \text{standard deviation}$ if the distribution is exactly exponential. Again, though, random sampling effects will destroy exact equality in just about every finite sample.

In both these cases you need to see if some type of "hypothesis testing" procedure has been developed for the types of equalities you want to test. If so, they are not part of standard textbooks, so you will need to do some research of the statistics literature. I suggest posting the question to the newsgroup "sci.stat.math" or similar on-line groups, or initial an on-line search through Google, for example. If you are close to a university you can go to the Statistics department and see of somebody can help you.

(3) Yes: $\lambda$ constant ==> homogeneous and $\lambda$ time-varying ==> non-homogeneous. However, from there on you are off-base. All you can say is that if $\lambda$ is constant then the expected number in t1, divided by t1, should equal the expected number in t2, divided by t2. Or, if t1 equals t2, the entire probability distributions of numbers in t1 and t2 are the same. Again, that says absolutely nothing about equality or non-equality of actual numbers in actual experiments or observations. Even if you had a genuine Poisson process and one observation was 800,000 units in some month, I would be truly shocked if another observation (for another month of exactly the same length) gave 800,000, or anything very close to it. For example, there is about a 91% chance that a month's total will lie outside the interval 800,000 ± 100 and about a 26% chance it will lie outside the interval 800,000 ± 1000.

Ah, I see. Now I can continue based on your remarks further. Thank you very much sir.