An Ostrowski's type inequality for a random variable whose probability density function belongs to \(L_\infty[a,b]\) (Q2702970)

Let \(X\) be a random variable with probability density function \(f\in L_{\infty }[a,b]\) and expectation \(E(X)\). The authors prove the following Ostrowski's type inequality NEWLINE\[NEWLINE \left|\Pr (X\leq x)-\frac{b-E(X)}{b-a}\right|\leq \left[ \frac{1}{4}+\frac{ \left( x-\frac{a+b}{2}\right) ^{2}}{(b-a)^{2}}\right] (b-a)\left\|f\right\|_{\infty },\forall x\in [a,b], NEWLINE\]NEWLINE where \(\|f\|_\infty= t\in [a,b]\), \(\sup f(t)<\infty \). The constant \(1/4\) is sharp. An application for a Beta random variable is also given.