13. Clean Plane (from 9th grade, 3 points). The vacuum robot "Orthogonal" decided to vacuum the coordinate plane. It starts from point $O$, located at the origin, and travels a straight distance $X_{1}$. At the point it arrives at, the vacuum turns and describes a circle centered at the origin. After completing the circle, it continues in the same direction and travels a straight distance $X_{2}$. Then it turns sideways to the origin again to describe another circle centered at the origin, and after completing it, travels a straight distance $X_{3}$. It continues to act in this manner. It is known that the values $X_{1}, X_{2}, X_{3}, \ldots$ follow some distribution with mean $a$ and variance $d$. Find the expected value of the area of the circle bounded by the $n$-th circle the vacuum has traveled.
Answer: $\pi n\left(d+a^{2}\right)$.

Solution. Exclude the circles from the vacuum cleaner's trajectory. It turns out that at each point it reaches after each step, the vacuum cleaner turns so that the direction of its movement forms a right angle with the direction to the origin (Fig. 7). Therefore, after the $n$-th step, the square of the distance $R$ from the vacuum cleaner to the origin can be found using the Pythagorean theorem:

$$
R^{2}=X_{1}^{2}+X_{2}^{2}+\ldots+X_{n}^{2}
$$

The area of the circle bounded by the $n$-th circle is

![](https://cdn.mathpix.com/cropped/2024_05_06_142b35c881fc974fcd25g-09.jpg?height=546&width=668&top_left_y=1800&top_left_x=1228)

Fig. 7

$$
S=\pi R^{2}=\pi\left(X_{1}^{2}+X_{2}^{2}+\ldots+X_{n}^{2}\right)
$$

Therefore,

$$
\mathrm{E} S=\pi\left(\mathrm{E} X_{1}^{2}+\ldots+\mathrm{E} X_{n}^{2}\right)=\pi n \mathrm{E} X_{1}^{2}=\pi n\left(\mathrm{D} X_{1}+\mathrm{E}^{2} X_{1}\right)=\pi n\left(d+a^{2}\right)
$$