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!
Best Answer
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 $\rho(x,y)=\frac{1}{4}.$