A Sharp Estimate for Probability Distributions

Abstract

We consider absolutely continuous probability distributions $f(x)dx$ on $\mathbb{R}_{\geq 0}$. A result of Feldheim and Feldheim shows, among other things, that if the distribution is not compactly supported, then there exist $z > 0$ such that most events in $\left\{X + Y \geq 2z\right\}$ are comprised of a 'small' term satisfying $\min(X,Y) \leq z$ and a 'large' term satisfying $\max(X,Y) \geq z$ (as opposed to two 'large' terms that are both larger than $z$) $$ \limsup_{z \rightarrow \infty}~ \mathbb{P}\left( \min(X,Y) \leq z | X+Y \geq 2z\right) = 1.$$ The result fails if the distribution is compactly supported. We prove $$\sup_{z > 0 } ~\mathbb{P}\left( \min(X,Y) \leq z | X+Y \geq 2z\right) \geq \frac{1}{24 + 8\log_2{( med(X) \|f\|_{L^{\infty}})}},$$ where $med(X)$ denotes the median. Interestingly, the logarithm is necessary and the result is sharp up to constants; we also discuss some open problems.