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Expected distance of a dart from center

EasyProbabilityRandom Variables~3m

A dart lands uniformly at random on a circular dartboard of radius 11. What is the expected distance from the center to where it lands?

Show hints (2)+
  1. Uniform over area: P(R≤r)=r2\mathbb{P}(R\le r)=r^2, so the density of RR is 2r2r.
  2. E[R]=∫01r⋅2r dr\mathbb{E}[R]=\int_0^1 r\cdot 2r\,dr.

Answer

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0.6667 (± 0.001)

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Asked at: Optiver, SIG

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