THE MEAN-MEDIAN-MODE INEQUALITY: COUNTEREXAMPLES
构造反例，证明当随机变量密度单峰且右偏时，众数≤中位数≤均值这一常见不等式不一定成立。
Let x be a random variable whose first three moments exist. If the density of x is unimodal and positively skewed, then counterexamples are provided which show that the inequality mode ≤ median ≤ mean does not necessarily hold.I thank Andrey Vasnev for help with the graphs and Jan Magnus for various helpful discussions. I also thank Martin Bland, Paolo Paruolo, Peter Phillips, Michael Rockinger, and a referee for their comments. ESRC grant R000239538 is gratefully acknowledged.