# Why are so many phenomena exponentially distibuted?

## Main Question or Discussion Point

http://en.wikipedia.org/wiki/Exponential_family

I can't count the number of times I've encountered exponential trends in my daily life:
- Productivity
- Skill differences
- Wealth differences
Not even all socioeconomic: You can even find it in nature:
- the energy amplitude of an electron decreasing exponentially as you get away from the center
- your mass increasing as you approach the speed of light
- the distribution of the length of phone calls to a call center...
- the salesvolume distribution of various products of a company
- the reasons of why certain products fail (Poisson charts)

There are many more examples (too many to count in other areas of the universe and life).

Why? What is the fundamental principle that is shared by all exponential trends?

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Char. Limit
Gold Member
They're all solutions of the profoundly simple differential equation:

dy/dx = y

i.e. that the rate of change depends on the amount of change.

thats a good way of looking at it but there is a more profound explanation that im missing. WHY does it drop less and less as the dependent variable gets bigger? why does the slope come closer and closer to 0. i read somewhere that it is due to some kind of causal relationship.

Ygggdrasil
Gold Member
I can't count the number of times I've encountered exponential trends in my daily life:
- Productivity
- Skill differences
- Wealth differences
[...]
- the distribution of the length of phone calls to a call center...
- the salesvolume distribution of various products of a company
Can't some of these be modeled by power law distributions as well?

thanks: thats what i actually meant.

however,my key questiosn are: why is the power law so prevalent? and why does any particular phenomenon obey that law?

D H
Staff Emeritus
thats a good way of looking at it but there is a more profound explanation that im missing. WHY does it drop less and less as the dependent variable gets bigger? why does the slope come closer and closer to 0. i read somewhere that it is due to some kind of causal relationship.
You get an exponential distribution for the time between events whenever the underlying events can happen at any point in time (i.e., continuous rather than discrete) and the events are independent from one another (i.e., the occurrence of an event neither excites nor inhibits the next occurrence).

There are lots of random processes that obey these two basic characteristics, and that is why you see so many exponential distributions.

oh so its mostly because you can have a set of arbitrary probabilities and then when they happen over and over again the result will be a skewed distribution ...even if the core probabilities are relatively close (.44^233 >> ..39^223)

thanks

Pengwuino
Gold Member
Another reason these kinds of distributions show up a lot is that many real world processes depend on this "rate of change is proportional to the current amount" relationship. For example, the rate of change of your bank account depends entirely on some constant (your interest rate) times the amount in the bank account. If you have a 1% interest rate per year, $100 will grow by$1 but if you have $10,000 dollars, the amount will grow by$100. If a factory has twice as many workers, it will typically produce twice as many goods, if you have twice as much of a radioactive element, it will give off twice as much radiation, etc etc.

i was just thinking about another phenomenon which i cant explain...

when i give fritz 1sec, 10sec, 100sec, 1000sec time to think about a move, the times where the recommendation differs fewer times among the time increments is exponentially more common than all four differing say.

0 difference 70%
1 difference 20%
2 difference 9%
3 difference 1%

but whats weird is you already account for the EXPTIME of seeking a move by the time increments (increasing in decades)...so what explains the fact that a power family remains in the frequency distribution?

Last edited:
apeiron
Gold Member
...my key questiosn are: why is the power law so prevalent? and why does any particular phenomenon obey that law?
This may be overkill for your purposes, but there is this excellent paper on the common patterns of nature - http://arxiv.org/abs/0906.3507

Each classic pattern was often discovered by a simple neutral generative model. The neutral patterns share a special characteristic: they describe the patterns of nature that follow from simple constraints on information. For example, any aggregation of processes that preserves information only about the mean and variance attracts to the Gaussian pattern; any aggregation that preserves information only about the mean attracts to the exponential pattern; any aggregation that preserves information only about the geometric mean attracts to the power law pattern.