  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 $f_X(x)$, this method
requires first setting a continuous, positive, and strictly decreasing function
$g_X(x)$ such that $-f_X(x)/g'_X(x)$ is a decreasing and increasing function,
$\forall x>x_0$, which results in upper and lower bounds, respectively, given
in the form $-f_X(x) g_X(x)/g'_X(x)$, $\forall x>x_0$, 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 $g_X(x)$ such that $f_X(x)/g'_X(x)$ is an increasing and
decreasing function, $\forall x<x_0$, which results in upper and lower bounds,
respectively, given in the form $f_X(x) g_X(x)/g'_X(x)$, $\forall x<x_0$. We
provide some examples of good candidates for the function $g_X(x)$. We also
establish connections between the new bounds and Markov's inequality and
Chernoff's bound. In addition, we provide an iterative method for obtaining
ever tighter lower and upper bounds, under certain conditions. Finally, we
provide numerical examples, where we show the tightness of these bounds, for
some chosen $g_X(x)$.
