305. A random variable $X$ is given by the density function $f(x)=0.5 x$ in the interval (0,2); outside this interval $f(x)=0$. Find the raw and central moments of the first, second, third, and fourth orders.
Solved and done. By the formula

$$
v_{k}=\int_{0}^{2} x^{k} f(x) \mathrm{d} x
$$

we find the initial moments:

$$
\begin{array}{ll}
v_{1}=\int_{0}^{2} x \cdot(0.5 x) \mathrm{d} x=\frac{4}{3} ; \quad v_{8}=\int_{0}^{2} x^{2} \cdot(0.5 x) \mathrm{d} x=2 \\
v_{3}=\int_{0}^{2} x^{3} \cdot(0.5 x) \mathrm{d} x=3.2 ; & v_{4}=\int_{0}^{2} x^{4} \cdot(0.5 x) \mathrm{d} x=\frac{16}{3}
\end{array}
$$

Let's find the central moments. The first-order central moment of any random variable $\mu_{i}=0$.

We will use formulas expressing central moments in terms of initial moments:

$$
\mu_{2}=v_{2}-v_{1}^{2} ; \mu_{3}=v_{3}-3 v_{1} v_{2}+2 v_{1}^{3} ; \mu_{4}=v_{4}-4 v_{1} v_{3}+6 v_{1}^{2} v_{2}-3 v_{1}^{4}
$$

Substituting the initial moments found earlier into these formulas, we get: $\mu_{2}=2 / 9, \mu_{3}=-8 / 135, \mu_{4}=16 / 135$.