Physical intuitions for simple statistical distributions

[schip666!]

I'm trying to understand why various statistical distributions are so common. For the most part, all I can find online is how to calculate and manipulate them... I did finally find a couple of refs that helped with Gaussians, this being one:
http://stat.ethz.ch/~stahel/lognormal/bioscience.pdf"

According to the above, a Gaussian Normal distribution arises due to having some "central tendency" or constraint summed with a buncha small plus/minus "random" variations -- such as in tossing darts at a target or playing Pachinko. And a Log Normal distribution is similar but the variations are multiplicative -- such as with exponential bacterial growth...Any other insights into these would be appreciated...

But the one I'm really interested in is the Poisson -- or its inverse, Exponential -- distribution which describes things like queuing behavior. For instance the time between emergency calls for a small volunteer fire department (and probably a large one too) -- I have exactly such data which matches the said distributions almost perfectly. But I have no intuition about how this happens. I would think that folks call 911 pretty much at random (modulo time of day and such) and that that would lead to a fairly even distribution across time. But no, I get that exponential instead. Why?

A secondary question, which I am just too stupid to be able to figure out on my own, is: Is the Exponential distribution actually a case of Log Normal? If so, then multiplicative variations would be an explanation, except I don't see a physical reason that queues might have that property.

I know, I know, I should post in Math. But this is a question about "Reality"...

 

[Bill_K]

Start with an event that has two possible outcomes, 'heads' and 'tails', or 'success' and 'failure', or whatever you want to call them. The probability of a success is p, while the probability of a failure is q, where p + q = 1.

The Binomial Distribution P(n) is the probability distribution of getting n successes out of N independent such events. It has a peak at n = Np, and an approximate width of √(Npq). Both the Gaussian and Poisson distributions are limiting cases of the Binomial Distribution.

You get the Gaussian Distribution by letting N get large (infinite, actually) in a way such that Np and Npq also get large. You then scale down this enormously large graph back to a reasonable size by switching to a variable x = (n - Np)/√(Npq), and look at P(x) - that's the Gaussian.

You get the Poisson Distribution P(n) by letting N get large and p get small, in such a way that a = Np stays finite. The Poisson Distribution is the probability distribution for a very large number of independent rare (p ≈ 0) events.

 

[schip666!]

Thanks for the quick answer...Of course, I have more questions...

First in notation.

Your second paragraph sets "n = Np", but the third has "x = (n - Np)/√(Npq)". Would not n - Np then always equal 0? Or are we presuming those values to be sets or something?

Then in the last paragraph "a = Np stays finite", should that not be n = Np ... I suppose a quibble, but since I'm still not sure what I'm looking at consistency is my hob-gob.

But the real question is... Given that Poisson is a distribution over very rare events, can I simulate/generate one using a bunch of "agents" who randomly (and very rarely) make entries in a queue? For instance, could I regenerate my real data, summarized here:
http://hondovfd.org/statistics.php"
Around the middle of that page is a graph of our Calls per Day compared to (what I hope to be) the ideal Poisson distrib.

I have about 5000 people in my fire district and about 500 emergency calls per year. That means about 1/10 of them need help every year (there are a number of repeat customers, but I hope we can ignore them here). So on any particular day each of those 5000 potential customers has a probability of instantiating of about (.1/365) == .00027.

Would I get the right result by stepping through days with 5000 guys having a .00027p of going "true" on each step? Or are there more subtleties to consider? Or I suppose I should just try it, eh?

 

[schip666!]

Well, I'll be dangnabbled... I did try it and it did work.
thanks!