Lack of well-defined average value A power-law has a well-defined mean over only if , and it has a finite variance only if ; most identified power laws in nature have exponents such that the mean is well-defined but the variance is not, implying they are capable of black swan behavior. This can be seen in the following thought experiment: imagine a room with your friends and estimate the average monthly income in the room. Now imagine the world's richest person entering the room, with a monthly income of about 1 billion US$. What happens to the average income in the room? Income is distributed according to a power-law known as the Pareto distribution (for example, the net worth of Americans is distributed according to a power law with an exponent of 2).
Lack of well-defined average value A power-law has a well-defined mean over only if , and it has a finite variance only if ; most identified power laws in nature have exponents such that the mean is well-defined but the variance is not, implying they are capable of black swan behavior. This can be seen in the following thought experiment: imagine a room with your friends and estimate the average monthly income in the room. Now imagine the world's richest person entering the room, with a monthly income of about 1 billion US$ What happens to the average income in the room Income is distributed according to a power-law known as the Pareto distribution (for example, the net worth of Americans is distributed according to a power law with an exponent of 2).