![Finding average squared distance of points in $[-1,1]\times[-1,1]$ to the origin](https://track.liberty.io/uploads/finding-average-squared-distance-of-points-in-1-1-times-1-1-to-the-origin_720.jpg)
$\begingroup$Let $R=[-1,1]\times [-1,1]$ be a square in $\mathbb R^2$. Find the average squared distance of points in $R$ to the origin.
My progress: I have learned how average value of function over $R$ is computed with the help of double integral. However, I can not understand the correlation of this problem with what I learned in multivariable calculus. I would be strongly appreciated if you could help me to explain what indeed the question asks to evaluate and why? Thanks in advance!
$\endgroup$1 Answer
$\begingroup$Assuming that you are looking for the expected value $E[D^2]$ of the squared Euclidean distance $D$ of the origin $(0,0)$ from points $r=(x,y)$ drawn from the uniform distribution on $R$, you need to compute the double integral
$$E[D^2]=\int_{-1}^1\int_{-1}^1\left(x^2+y^2\right) \rho(x,y) dx dy $$ where the uniform area density is $\rho(x,y)=\frac{1}{4}.$
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