NOTE ON DISPERSION OF $X^\alpha$

D. Banjevi\'c and D. Braticevi\'c

Abstract: 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.

Keywords: nonnegative random variables, inequalities for moments, unbias estimation, binomial distribution

Classification (MSC2000): 60E15, 62F10; 62F11, 62F12

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