Example 6. Find the initial moments of the random variable $X$ with the probability density function

$$
p(x)= \begin{cases}0 & \text { if } x \leq 1 \\ \frac{5}{x^{6}} & \text { if } x>1\end{cases}
$$
Solution. According to formula (2.6.6), we obtain

$$
v_{k}=M\left(X^{k}\right)=\int_{-\infty}^{+\infty} x^{k} \frac{5}{x^{6}} d x=5 \int_{1}^{+\infty} x^{k-6} d x=\left.5 \frac{x^{k-5}}{k-5}\right|_{1} ^{+\infty}
$$

The improper integral converges for $k<5$. In this case,

$$
v_{k}=M\left(X^{k}\right)=\left.5 \frac{x^{k-5}}{k-5}\right|_{1} ^{+\infty}=\frac{5}{5-k}
$$

Therefore, the considered random variable $X$ has moments of the first, second, third, and fourth orders, but does not have moments of the fifth and higher orders.