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!

 

[pmacias]

Dear reader,

Thank you for your interest in understanding the commonality of various statistical distributions. I can provide some insights into this topic.

Firstly, it is important to recognize that statistical distributions are not just theoretical constructs, but they arise from real-world phenomena and observations. This means that there is a physical reason behind the patterns we see in these distributions.

Let's start with the Gaussian or Normal distribution. This distribution is often observed in natural phenomena because of the central limit theorem. This theorem states that when independent random variables are added together, their sum tends to follow a Gaussian distribution. This is because in many natural processes, there are multiple factors contributing to the overall outcome, and these factors are often independent and randomly distributed. For example, in the case of tossing darts at a target, the small variations in throwing angle, force, and other factors can be seen as independent and random, leading to a Gaussian distribution of the final outcome.

The Log Normal distribution, as you mentioned, is similar to the Gaussian but with multiplicative variations instead of additive ones. This can be seen in processes such as exponential growth, where small variations in growth rate can lead to large differences in the final outcome.

Now, let's move on to the Poisson and Exponential distributions. These distributions are commonly observed in queuing behavior, as you mentioned. The reason for this is due to the nature of queuing systems. In a queuing system, there is a constant arrival rate of customers, but the service time for each customer is variable and can follow any distribution. This leads to an Exponential distribution for the time between arrivals, as the time between arrivals is independent of the previous one. The Poisson distribution, on the other hand, describes the number of arrivals in a given time period, and it arises from the assumption that arrivals follow a Poisson process, where there is a constant arrival rate and arrivals are independent of each other.

To answer your secondary question, the Exponential distribution is not a case of the Log Normal distribution. While both distributions have a similar shape, they arise from different underlying processes. The Exponential distribution arises from the constant arrival rate in queuing systems, whereas the Log Normal distribution arises from multiplicative variations in processes such as exponential growth.

I hope this provides some insight into the physical reasons behind the commonality of these distributions. It is important to note that these are just a few examples, and there are many other statistical distributions that arise