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?