Sharp inequalities of Bienaymé-Chebyshev and Gauß type for possibly asymmetric intervals around the mean (Q6051852)

The authors establish sharp inequalities on probabilities of the form \(P(Z\leq -u\mbox{ or }Z\geq v)\) for random variables \(Z\) (typically assumed to have zero mean and unit variance) and some \(u,v>0\). Inequalities are derived without any further assumptions on \(Z\), and in the following cases where some further restrictions are imposed: \begin{itemize} \item the distribution of \(Z\) is symmetric; \item the distribution function of \(Z\) is concave on \([0,\infty)\), and zero on \((-\infty,0)\); \item \(Z\) is unimodal; \item \(Z\) is unimodal with mean and mode coinciding; and \item \(Z\) is unimodal with a symmetric distribution. \end{itemize} In each case sharp upper bounds are established, with emphasis given to highlighting the situations in which these bounds are achieved. Results presented here unify and generalize well-known inequalities due to Gauss, Bienaymé, Chebyshev and Cantelli, among others.