Example 10. Find the numerical characteristics $M(X), D(\dot{X}), \sigma(X)$ of a continuous random variable $X$ given by the probability density function

$$
p(x)= \begin{cases}0 & \text { if } x \leq 0 \\ 2 x & \text { if } 0 < x \leq 1 \\ 0 & \text { if } x > 1\end{cases}
$$
Solution. Using formula (2.4.7), we find the expected value:

$$
M(X)=\int_{0}^{1} 2 x \cdot x d x=\int_{0}^{1} 2 x^{2} d x=\left.\frac{2}{3} \cdot x^{3}\right|_{0} ^{1}=\frac{2}{3}
$$

Using formulas (2.5.13) and (2.5.16) respectively, we obtain

$$
\begin{gathered}
D(X)=\int_{0}^{1}\left(x-\frac{2}{3}\right)^{2} \cdot 2 x d x=\int_{0}^{1}\left(x^{2}-\frac{4}{3} x+\frac{4}{9}\right) 2 x d x=\int_{0}^{1}\left(2 x^{3}-\frac{8}{3} x^{2}+\frac{8}{9} x\right) d x= \\
=\left.\left(\frac{x^{4}}{2}-\frac{8 x^{3}}{9}+\frac{4}{9} x^{2}\right)\right|_{0} ^{1}=\frac{1}{2}-\frac{8}{9}+\frac{4}{9}=\frac{1}{18} \\
D(X)=\frac{1}{18}, \quad \sigma(X)=\sqrt{1 / 18} \approx 0.24
\end{gathered}
$$