1751. Find the expression for the third-order central moment in terms of the initial moments.
Solution. $\quad \mu_{3}=\overline{(X-\bar{X})^{3}}=\overline{X^{3}-3 X^{2} \bar{X}+3 X(\bar{X})^{2}-(\bar{X})^{3}}=$

$$
=\bar{X}^{3}-3 \overline{X^{2}} \bar{X}+3 \bar{X}(\bar{X})^{2}-(\bar{X})^{3}=v_{3}-3 v_{1} v_{2}+2 v_{1}^{3}
$$