How to prove the earning decomposition of 2 people in mediocristan and extremistan? In his book The Black Swan (chapter 15, section The Mandelbrotian), Nassim Nicholas Tayeb says that if the sum of the earnings of 2 people is 1 million, the most probable decomposition in Mediocristan is 500,000 each and in Extremistan, it is 950,000 / 50,000.
Medicristan means normal law, Extremistan means power law.
So he says that if the earnings followed a normal law, the most probable decomposition of a random sample of 2 whose sum is 1 million is (500,000 ; 500,000). The earnings follow a power law (the parameters are not precised), so the most probable decomposition of a random sample of 2 whose sum is 1 million is (950,000 ; 50,000).
Is he right?
If yes, how do you prove it?
 A: The point is that the normal distribution falls off much faster than a power law, because the exponent is squared.  It is hard to say what would be reasonable parameters for a normal distribution of incomes-it is clearly far from that.  The 2010 median in the US was about 26k and 94% were below 100k.  If we try to take those for parameters of a normal distribution we would say the mean was 26k and the standard deviation about 37k.  That would say even 250k was six sigma above the mean and should be only one person in a billion-clearly badly wrong.
Trying to approach the remark about the million dollar households, let us stretch the distribution to a mean of 0 (it doesn't matter) and a standard deviation of 100k.  Then the million dollar household is either two at +5 sigma, probability one in $10^{-12}$ or one at +10 sigma, probability $10^{-33}$
In the power law, the distribution is $\frac {a-1}{x_{min}}(\frac x{x_{min}})^{-a}$.  We need to cut off at $x_{min}$ to avoid the infinity at zero.  Based on the fact that 6% of individuals are above 100k and taking 10k as the minimum, we get $\frac {1.15}{10,000}(\frac x{x_{min}})^{-2.15}$ for the distribution.  Then the probability of two 500k's is $(0.011)^2=0.00012$ while the probability of a single 1M is $0.005$  It seems an even bigger thing is that in Medicristan there are no millionaires.
This argument assumes there is no correlation between the incomes of a couple.  We also know that is not true.
