# Why are so many phenomena exponentially distibuted?

1. Jan 29, 2012

### Helicobacter

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?

2. Jan 30, 2012

### Char. Limit

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.

3. Jan 30, 2012

### Helicobacter

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.

4. Jan 30, 2012

### Ygggdrasil

Can't some of these be modeled by power law distributions as well?

5. Feb 20, 2012

### Helicobacter

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?

6. Feb 20, 2012

### D H

Staff Emeritus
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.

7. Feb 20, 2012

### Helicobacter

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

8. Feb 20, 2012

### Pengwuino

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.

9. Feb 20, 2012

### Helicobacter

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: Feb 20, 2012
10. Feb 20, 2012

### apeiron

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