Why are so many phenomena exponentially distibuted?

  • Thread starter Helicobacter
  • Start date
  • Tags
    Phenomena
In summary, exponential trends are exhibited by many different processes in many different areas of life. There is a fundamental principle that is shared by all exponential trends: the rate of change depends on the amount of change.
  • #1
Helicobacter
158
0
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
or in business:
- 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?
 
Physics news on Phys.org
  • #2
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
thats a good way of looking at it but there is a more profound explanation that I am 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
Helicobacter said:
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?
 
  • #5
thanks: that's 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
Helicobacter said:
thats a good way of looking at it but there is a more profound explanation that I am 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.
 
  • #7
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
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
i was just thinking about another phenomenon which i can't 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 what's 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:
  • #10
Helicobacter said:
...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.
 

FAQ: Why are so many phenomena exponentially distibuted?

1. Why are exponential distributions commonly observed in nature?

Exponential distributions are commonly observed in nature because they arise from processes that involve random events occurring over time, such as radioactive decay, disease spread, and financial market fluctuations. These processes are characterized by a constant rate of occurrence, which results in an exponential distribution.

2. What is the relationship between exponential distributions and the law of large numbers?

The law of large numbers states that as the sample size increases, the sample mean approaches the true population mean. Exponential distributions follow this law because as the sample size increases, the frequency of rare events decreases, resulting in a distribution that is closer to the expected mean.

3. Can exponential distributions be used to model non-random phenomena?

Yes, exponential distributions can be used to model non-random phenomena by adjusting the parameters of the distribution. For example, in queueing theory, the arrival rate of customers at a service station can be modeled using an exponential distribution with a constant arrival rate.

4. What is the difference between an exponential distribution and a normal distribution?

An exponential distribution is a probability distribution that describes the time between events in a Poisson process. It is a right-skewed distribution with a long tail, while a normal distribution is symmetrical and bell-shaped. Additionally, the values in an exponential distribution can only be positive, while a normal distribution can have both positive and negative values.

5. How can we use exponential distributions to make predictions?

Exponential distributions can be used to make predictions by calculating probabilities of events occurring within a specific time frame. For example, if we know the average time between earthquakes in a certain region is 10 years, we can use an exponential distribution to predict the likelihood of an earthquake occurring within the next 5 years.

Similar threads

Replies
1
Views
3K
Replies
28
Views
10K
Replies
61
Views
15K
Replies
16
Views
8K
Replies
96
Views
7K
Replies
5
Views
2K
Replies
1
Views
2K
Back
Top