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If a dart thrower had the ability to throw a dart at a dart board (with radius $r$) and hit it every time in a random position, what would the probability density function of the distances from where the dart landed on the board to the centre?

My friend and I were arguing about what the shape of the graph would be. My friend says it would be a horizontal line, while I say otherwise. Who is correct and how would a problem like this be approached?

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    $\begingroup$ The answer depends on the distribution of the darts on the board. Would you be happy to assume the darts would land uniformly over the board? $\endgroup$ Commented May 17, 2020 at 12:42
  • $\begingroup$ Yes, the land uniformly over the board $\endgroup$ Commented May 17, 2020 at 12:44

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We can assume the radius of the circle is $1.$ Let $R$ be a random variable denoting the distance between the center and a dart landing on the board. For any $r\in [0,1]$, the probability a dart lands within $r$ of the center is equal to the ratio of the area of the circle of radius $r$ to the area of the circle of radius $1.$ Thus we have $\mathbb{P}(R \leq r) = \displaystyle \frac{ \pi r^2}{\pi} = r^2$ so the probability density function is $p_R(r) = 2r$ for $r\in [0,1]$ and $0$ elsewhere.

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