1. A point is chosen randomly with uniform distribution in the interior of a circle of radius
1. What is its expected distance from the center of the circle?
Solution:
$2 / 3$
The probability of the point falling between a distance $r$ and $r+d r$ from the center is the ratio of the area of the corresponding annulus to the area of the whole circle: $\frac{\pi\left[(r+d r)^{2}-r^{2}\right]}{\pi} \rightarrow \frac{2 \pi r d r}{\pi}=2 r d r$ for small values of $d r$. Then the expected distance is $\int_{0}^{1} r \cot 2 r d r=\frac{2}{3}$.