Theory of probability distributions | Statistical inequalities | Statistical deviation and dispersion

Popoviciu's inequality on variances

In probability theory, Popoviciu's inequality, named after Tiberiu Popoviciu, is an upper bound on the variance σ2 of any bounded probability distribution. Let M and m be upper and lower bounds on the values of any random variable with a particular probability distribution. Then Popoviciu's inequality states: This equality holds precisely when half of the probability is concentrated at each of the two bounds. Sharma et al. have sharpened Popoviciu's inequality: Popoviciu's inequality is weaker than the Bhatia–Davis inequality which states where μ is the expectation of the random variable. In the case of an independent sample of n observations from a bounded probability distribution, the gives a lower bound to the variance of the sample mean: (Wikipedia).

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Related pages

Inequality of arithmetic and geometric means | Tiberiu Popoviciu | Variance | Random variable | Bhatia–Davis inequality | Probability theory | Probability distribution