TITLE: Difference between power law distribution and exponential decay
QUESTION [36 upvotes]: This is probably a silly one, I've read in Wikipedia about power law and exponential decay. I really don't see any difference between them. For example, if I have a histogram or a plot that looks like the one in the Power law article, which is the same as the one for $e^{-x}$, how should I refer to it?

REPLY [7 votes]: If there is anybody landing on this from Nassim Nicholas Taleb's The Black Swan the issue at stake is how doubling a random variable affects the probability in power law distributions as opposed to a normal or Gaussian distribution.
In the case of continuous random variables, the pdf illustrates the difference:
A power law distribution has a pdf of the form
$$f_X(x) =C x^{-\alpha},$$
where $C$ is a constant, and $\alpha$ is the law's exponent.
The effect of doubling $x$ will remain constant across the domain: For example, the ratio in the pdf between people who make $\$50,000$ per year to those that make $\$25,000$ will be the same as the ratio between people that make $\$10,000,000$ to those that make $\$5,000,000:$
$$\frac{(2x)^{-\alpha}}{x^{-\alpha}}=2^{-\alpha}$$
an attribute called scale invariance.
This is in contrast to the rapid decay in the normal distribution, which in the standardized form has the following pdf:
$$f_X(x)=\frac{1}{\sqrt{2\pi}}\exp\left( -\frac{x^2}{2}\right)$$
Doubling the value of $x,$ amounts to raising to the third power the exponential (un-normalized) part of the pdf:
$$\begin{align}
\frac{f_X(2x)}{f_X(x)}&=\exp\left(-\frac{(2x)^2}{2} +\frac{{}x^2}{2}\right)\\[2ex]
&=\exp\left( -\frac{3x^2}{2}\right)\\[2ex]
&=\left(\exp\left(-\frac{x^2}{2} \right)\right)^{3}\\[2ex]
&=\frac{1}{\left(\mathrm e^{x^2/2}\right)^3}
\end{align}$$
which can be visually plotted as:

Therefore, at asymptotic values the relative probability between an extreme event and that of its value doubled will rapidly tend to zero, whereas in a power law it will remain stubbornly constant.