280. Find the mathematical expectation of the random variable $X$, given by the distribution function

$$
F(x)=\left\{\begin{array}{ccr}
0 & \text { for } & x \leqslant 0 \\
x / 4 & \text { for } & 0 < x \leqslant 4 \\
1 & \text { for } & x > 4
\end{array}\right.
$$
Solution. Find the density function of the random variable $\boldsymbol{X}$:

$$
f(x)=F^{\prime}(x)=\left\{\begin{array}{ccc}
0 & \text { for } & x4 .
\end{array}\right.
$$

Find the required expected value:

$$
M(X)=\int_{0}^{4} x f \cdot(x) \mathrm{d} x=\int_{0}^{4} x \cdot(1 / 4) \mathrm{d} x=2
$$