Let \(F(x)\) be the distribution function of \(\xi\). Prove that if \(\mathbf{M} \xi\) exists, then

\[ M \xi=\int_{0}^{\infty}[1-F(x)+F(-x)] d x \]

and for the existence of \(\boldsymbol{M} \xi\) it is necessary that \[ \lim _{x \rightarrow-\infty} x F(x)=\lim _{x \rightarrow \infty} x[1--F(x)]=0 \]