Mean distance of a random dart from the centre

Question

A dart lands at a point chosen uniformly over a circular board of radius $1$ . What is the expected distance from the landing point to the centre of the board?

Short answer

$2/3$ . The distance $R$ satisfies $P(R\le r)=r^2$ , so its density is $2r$ on $[0,1]$ and $E[R]=\int_0^1 2r^2\,dr=\tfrac23$ . Points near the rim dominate because there is more area there.

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Lessons behind this question

Continuous Random Variables and Probability Density Functions
Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited

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