Some inequalities for random variables whose probability density functions are bounded using a pre-Grüss inequality (Q2718044)

Let \(X\) be a random variable having the probability density function \( f:[a,b]\rightarrow \Re \) , the cumulative distribution function \( F(x)=\int_{a}^{b}f(t)dt\) , and the expectation \(E(X)=\int_{a}^{b}tf(t)dt\) . In this paper it is proved that if there exist the constants \(\gamma ,\varphi \) such that \(0\leq \gamma \leq f(t)\leq \varphi \leq 1\) for a.e. \(t\) on \([a,b]\) , then we have the inequality NEWLINE\[NEWLINE \left|E(X)+(b-a)F(x)-x-\frac{b-a}{2}\right|\leq \frac{1}{4\sqrt{3}} (\varphi -\gamma)(b-a)^{2} NEWLINE\]NEWLINE for all \(x\in [a,b]\) . Some related results involving \(p-\)moments of \(X\) are also proved.