I wrote this note primarily for myself to understand the equation (6) on the page 5 from Nassim Nicholas Taleb’s paper On the Statistical Differences between Binary Forecasts and Real World Payoffs:
(1) 
where:
- x is a random variable,
- g(x) is the probability density function (PDF) of x, which is Gaussian distribution (centered and normalized).
The author assumes familiarity with the equation 1 and, therefore, doesn’t elaborate on it. The steps below show where this equation came from.
More generally, the expected payoff I for all values of x above K, i.e. the tail of the distribution, is:
(2) 
where f(x) is the payoff function.
Normally, we would use integration by parts for calculating the integral above.
But the author considers a special case where the payoff function f(x) is linear:
(3) 
So, 2 becomes:
(4) 
The integral 4 is much easier to solve than more general 2, because the first derivative of the centered and normalized Gaussian PDF has the following property:
(5) 
We will integrate the equation and apply the second fundamental theorem of calculus to the left side of 5:
(6) 
Notice that the right side became our expected payoff function, therefore:
(7) 
Since the probability approaches 0 as x approaches infinity, the expected payoff of all x above K equals the probability of K:
(8) 
