Inequalities for the median of the gamma distribution
The function \(F(x)=\frac{1}{n!}\int_0^xt^n\mathrm{e}^{-t}dt\), \(x\geq 0\), is a~distribution function of the \(\Gamma(n+1,1)\) distribution. Denote by \(\lambda_n\) its median, i.e. the unique solution of the equation \(F(x)=\frac{1}{2}\). Various inequalities establishing sharp lower and upper bounds of some expressions involving \(\lambda_n\) are given. Among others it is proved that for all positive integers \(n\) the inequality \(\frac{\alpha}{n(n+1)}<\frac{\lambda_n}{n}-\frac{\lambda_{n+1}}{n+1}\leq\frac{\beta}{n(n+1)}\) holds with the best possible constants \(\alpha=\frac{2}{3}\) and \(\beta=2\lambda_1-\lambda_2\approx 0.6826\). This is only one example of many inequalities of this type proved in the paper.
scientific article; zbMATH DE number 5608842