A Method For Bounding Tail Probabilities

arXiv:2402.13662v3 Announce Type: replace-cross
Abstract: We present a method for upper and lower bounding the right and the left tail probabilities of continuous random variables (RVs). For the right tail probability of RV $X$ with probability density function $f (x)$, this method requires first setting a continuous, positive, and strictly decreasing function $g (x)$ such that $-f (x)/g’ (x)$ is a decreasing and increasing function, $forall x>x_0$, which results in upper and lower bounds, respectively, given in the form $-f (x) g (x)/g’ (x)$, $forall x>x_0$, where $x_0$ is some point. Similarly, for the upper and lower bounds on the left tail probability of $X$, this method requires first setting a continuous, positive, and strictly increasing function $g (x)$ such that $f (x)/g’ (x)$ is an increasing and decreasing function, $forall x