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I have two points say $\mathbf{p_1} = (x_1, y_1)$ and $\mathbf{p_2}= (x_2, y_2)$ which are uniformally distributed with parameter 0, 1, i.e., $\mathcal{U}(0, 1)$. These two points are generated inside a circle of radius $R=20$.

I am interesting in finding the mean and the distribution of the Euclidean distance between $\mathbf{p_1}$ and $\mathbf{p_2}$.

Let $d_{12}$ be the distance between $\mathbf{p_1}$ and $\mathbf{p_2}$.

$$d_{12} = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2},$$

What is $\mathbb{E}[d_{12}]$?

What I did is the following:

Let $X=x_1-x_2$ and $Y=y_1-y_2$ then

$$f_X(x) = 1-x+2\cdot x\cdot H(-x),$$ and $$f_Y(y) = 1-y+2\cdot y\cdot H(-y),$$ where $H(\cdot)$ is the Heaviside step function.

How can do the rest?

drzbir
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  • These single variable distributions won't work for calculating this, since x and y are correlated. You need a joint distribution – Paul Jun 20 '15 at 19:30
  • See Wikipedia on the Rayliegh distribution. – BruceET Jun 21 '15 at 03:34
  • Related: http://math.stackexchange.com/questions/135766/average-distance-between-two-points-in-a-circular-disk and http://math.stackexchange.com/questions/22937/finding-the-mean-distance-between-n-points-evenly-distributed-in-a-disc-of-radiu – Henry Sep 02 '15 at 22:36

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