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 $g_X(x)$.

翻译：我们提出了一种连续随机变量右尾和左尾概率上界与下界的界定方法。对于概率密度函数为 \(f_X(x)\) 的随机变量 \(X\) 的右尾概率，该方法要求首先设定一个连续、正且严格递减的函数 \(g_X(x)\)，使得 \(-f_X(x)/g'_X(x)\) 在 \(\forall x > x_0\) 时分别为递减和递增函数，从而分别得到形如 \(-f_X(x) g_X(x)/g'_X(x)\) 的上界和下界，其中 \(x_0\) 为某一点。类似地，对于 \(X\) 的左尾概率的上界和下界，该方法要求首先设定一个连续、正且严格递增的函数 \(g_X(x)\)，使得 \(f_X(x)/g'_X(x)\) 在 \(\forall x < x_0\) 时分别为递增和递减函数，从而分别得到形如 \(f_X(x) g_X(x)/g'_X(x)\) 的上界和下界。我们给出了函数 \(g_X(x)\) 的一些良好候选实例。同时，我们建立了新边界与马尔可夫不等式及切尔诺夫界之间的联系。此外，在特定条件下，我们提供了一种迭代方法以获得更紧的下界和上界。最后，通过数值示例展示了针对某些选定的 \(g_X(x)\) 所得边界的紧致性。