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

A point is chosen uniformly at random from the area of a disk of radius one. What is its expected distance from the center? ### Constraints and Clarifications "Uniform" means equal-area regions 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` is an area inside the original disk. Its probability follows from the ratio of those areas. ``` ### What a Strong Answer Covers - The radial cumulative distribution implied by uniform area sampling. - A correct density or tail-integral calculation of the expected radius. - An explanation of why a uniform radius gives a different distribution of points. - The scaling of the answer for a disk with a different radius. ### Follow-up Questions 1. What expected distance would result from choosing the radius uniformly between zero and one instead? 2. How would you generate a uniform-area point using independent uniform random variables for the angle and radial calculation?