This article is cited in 4 papers

Short Communications

An inequality for moments of a random variable

V. V. Petrov

Leningrad

Abstract: Let $X$ be a random variable. For any $r>0$, we set $\beta_r=\mathbf E|X|^r$. The following inequality is proved:
$$ \beta_r^{1/r}\le\gamma^{1/r-1/s}\beta_s^{1/s}\quad(r<s) $$
where $\gamma=\mathbf P(X\ne0)$. This inequality is optimal in a certain sense.

Received: 11.11.1974

 English version: Theory of Probability and its Applications, 1976, 20:2, 391–392

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