Find the Expected Distance of a Uniform Point from a Disk's Center

A point is selected uniformly at random from the interior of a disk of radius one. What is the expected value of its distance from the center?

Constraints and Clarifications

"Uniform" means regions of equal area have equal probability. It does not mean choosing the radius uniformly. Let $R$ be the point's distance from the center, and derive its distribution before computing the expectation.

Hint: Compare the inner disk with the whole disk

The event that the distance is at most $r$ corresponds to a region inside the original disk. Its probability is given by the ratio of those areas.

What a Strong Answer Covers

• The cumulative distribution of the radius that follows from sampling uniformly by area.
• A correct computation of the expected radius using either the density or a tail integral.
• Why choosing the radius uniformly produces a different spatial distribution of points.
• How the result scales when the disk has a different radius.

Follow-up Questions

1. If the radius were instead chosen uniformly between 0 and 1, what expected distance would that produce?
2. How can you generate a point uniformly by area using independent uniform random variables for the angle and for the radial calculation?

Overview: Derive the radial distribution and the expected distance for a point sampled uniformly by area within a unit disk.