On some inequalities for the expectation and variance
Let \(f:[a,b]\rightarrow\mathbb{R}\) be the p.d.f. of the random variable \(T\) and \(M_{n}=\int_{a}^{b}t^{n}f(t) dt\) be the moment of order \(n\) of \(T\). As usual, \(E(T)=M_{1}\) denotes the expectation of \(T\) and \(\sigma ^{2}(T)=M_{2}-M_{1}^{2}\) \ its variance. The authors obtain a variety of identities and bounds of the expression \(\sigma ^{2}(T)+[E(T)-\alpha ][E(T)-\beta ]\) , where \(\alpha ,\beta \in [a,b]\) .
scientific article; zbMATH DE number 1625580