Let $Y = (Y_1,Y_2,\ldots , Y_m)$ be a random vector with uniform distribution on the cube ${[- \sqrt 3, \sqrt3]}^m$.
1) Prove that the Y_i are mutually independent.
2) What is $E(Y_i)$, $E(Y_i^2)$, $Var(Y_i^2)$?
3) Let $\left\lVert Y\right\rVert$ = $\sqrt {\sum_{i=1}^m Y_i^2}$ represent distance from Y to origin. Verify that $$E(|\left\lVert Y\right\rVert - \sqrt m|^2)<1,$$ then verify $$P(\left\lVert Y\right\rVert-\sqrt m>s)>\frac 1 {s^2}$$ for $s>0$.
For 2) So I think Y_i should equal to $E(Y_1)+...+E(Y_m)$ since it is unfiorm distribution buthow do I find $E(Y_1)$? and what do i do for $E(Y_1^2)$ do I just do $E(Y_1)^2$ since they should be independent? but I do not know how to prove 1). Any hint would be appreciated thanks.
