Example 11. Find the mathematical expectation of the random variable $X$, if the distribution function of this variable is known

$$
F(x)= \begin{cases}0 & \text { if } x \leq 0 \\ x^{2} & \text { if } 0 < x \leq 1\end{cases}
$$
Solution. First, let's find the probability density function of this variable. Using formula (2.3.5), we get

$$
p(x)= \begin{cases}0 & \text { if } x \leq 0 \\ 2 x & \text { if } 0 < x \leq 1 \\ 0 & \text { if } x > 1\end{cases}
$$

Therefore,

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