The exceedance probability PK, as a random variable bounded between 0 and 1, necessarily has thin tails with finite mean, variance and all moments.

This is a mathematical claim, judged as a statement of probability theory. True: standard probability theory confirms that a random variable whose support lies within [0,1] has a finite mean, a finite variance and finite moments of all orders, and has no heavy (power-law or otherwise unbounded) tails, since its tail probabilities are zero beyond the bounds. Mostly true: the moments part holds, but 'thin tails' is shown to be an imprecise label for bounded support, for example because a bounded variable can still pile its mass near the boundaries. Mixed: one of the two parts (all moments finite; thin tails) holds and the other does not. Mostly false: the result holds only under extra conditions the claim leaves out. False: a valid counterexample exists, meaning a random variable bounded in [0,1] with an infinite or undefined moment.

Set before the research began.

1. E1 4.4 Moments - Probability and Statistics jjacobs.me (Probability and Statistics textbook, ch. 4) · 14 October 2026

   “If the random variable X is bounded, that is, if there are finite numbers a and b such that Pr(a ≤ X ≤ b) = 1, then all moments of X must necessarily exist.”

2. E2 CS5275 Lecture 2: Concentration Inequalities National University of Singapore (Jonathan Scarlett, CS5275 course notes) · 20 January 2025

   “Claim: Any bounded random variable Z ∈ [a, b] has variance at most Var[Z] ≤ (b−a)2”

A probability textbook [E1] says that a bounded random variable must have all of its moments, so a PK confined to [0,1] has a finite mean, a finite variance and finite higher moments. University course notes [E2] state that any bounded random variable has a bounded, finite variance. The evidence supports the finite-moments part of the claim and gives no counterexample. Neither quote uses the term 'thin tails', but E1's definition of a bounded variable (Pr(a ≤ X ≤ b) = 1) means its tail probabilities beyond the bounds are zero, which is how the criteria define having no heavy tails.