Tail Probability Expectation: Decomposing Expected Impact Above a Threshold

Summary

The document asks how to interpret a tail expectation for a nonnegative random variable when impact grows linearly with the variable. It distinguishes the probability of exceeding a threshold from the expected impact contributed by outcomes above that threshold. The latter is an integral of the variable times its density, so it reflects both how often the tail occurs and how large tail outcomes are.

For finite first moments, the stated tail formula decomposes that expectation into the threshold multiplied by its exceedance probability, plus the integral of exceedance probabilities over higher thresholds. This is a tail-integral identity: the variable does not appear explicitly in the second term because each probability already measures the portion of the distribution above its integration level. The document raises the question but does not include an answer or worked derivation, and its setup is limited to a nonnegative variable with finite mean and linear impact.

Key ideas

• Exceedance probability measures how often a variable crosses a threshold, while tail expectation also captures the size of exceedances.
• For linear impact, the tail expectation can be written as the threshold times its exceedance probability plus an integral of higher-threshold probabilities.
• The tail-integral identity assumes a nonnegative random variable with a finite first moment.
• The document poses the derivation question but supplies no worked explanation.

Tags

# Tail Probability Expectation Formula


# Tail Probability Expectation Formula












On page one of the following paper: The Probability Conflation: A Reply by Nassim Nicholas Taleb et al.,

the following calculation is made:

Begin Quoted Passage

$K \in R^+$ is a threshold f(.) density function for random variable $X \in R^+$ $P_K = P(X>K) \in [0,1]$ the probability of exceeding it $g(x): R^+ \rightarrow R$ an impact function

$G_K = \int_K^{\infty} g(x)f(x)dx (1)$

The complimentary CDF function at K:

$P_K = \int_K^{\infty} f(x) dx(2)$

The error comes from conflating the properties of GK which those of PK, often associating PK with some constant representing the presumed impact associated with the threshold K

The intuition of the difference can be shown as follows: assuming g(x) = x, for X a random variable with finite first moment, we have, focusing on the positive domain, generalizing the Tail Probability Expectation Formula,

$G_K = KP_K + \int_K^{\infty} P_x dx$

End Quoted Passage

The first term is straightforward application of equation 1 with $g(x)=x$ at the start of the integral, is this correct?

I am confused as to why x does not appear in the second term, can someone please explain?

$G_K = \int_K^{\infty} xf(x)dx$ $\neq$ $\int_K^{\infty} f(x) dx$ $=$ $\int_K^{\infty} P_x dx$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.