Note on Dispersion of Xalpha


D. Banjević, D. Bratičević


Some inequalities for moments of $X^\alpha$, $0<\alpha\leq 1$, $X$ nonneg. r. v., are presented, for example $DX^\alpha\leq (DX)^\alpha$, $DX^\alpha\leq (DX)/(EX)^{2(1-\alpha)}$, $$ (EX)^\alpha-EX^\alpha \leq (1-\alpha)(DX)/(EX)^{2-\alpha}. $$ It is proved that $nD\overline X_n^\alpha\to\alpha^2(DX)/(EX)^{2(1-\alpha)}$, $n\to\infty$, where $X_1,X_2,\dots,X_n$ are i. \i d. r. v. and $\overline X_n= (X_1+X_2+\dots+ X_n)/n$. The estimation of $\sqrt{EX}$ is considered, and for binominal case some numerical evaluations are given.