The convex transform order is one way to make precise comparison between the skewness of probability distributions on the real line. We establish a simple and complete characterization of when one Beta distribution is smaller than another according to the convex transform order. As an application, we derive monotonicity properties for the probability of Beta distributed random variables exceeding the mean or mode of their distribution. Moreover, we obtain a simple alternative proof of the mode-median-mean inequality for unimodal distributions that are skewed in a sense made precise by the convex transform order. This new proof also gives an analogous inequality for the anti-mode of distributions that have a unique anti-mode. Such inequalities for Beta distributions follow as special cases. Finally, some consequences for the values of distribution functions of binomial distributions near to their means are mentioned.

中文翻译：

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具有某些后果的 Beta 分布的凸变换顺序

凸变换阶数是在实线上对概率分布的偏度进行精确比较的一种方法。我们根据凸变换顺序建立了一个 Beta 分布何时小于另一个的简单而完整的表征。作为一个应用，我们推导出 Beta 分布随机变量超过其分布均值或众数的概率的单调性。此外，我们获得了单峰分布的模式-中值-均值不等式的一个简单的替代证明，这些分布在某种意义上被凸变换阶变得精确。这个新的证明还给出了具有唯一反模式的分布的反模式的类似不等式。Beta 分布的这种不等式遵循特殊情况。最后，