expectation of uniformly distributed $n$ number of samples

Question

I am trying to fine the expectation: $E((x_1+ x_2+ \cdots +x_n )^2)$ as a function of $n$ where all $x_1$ to $x_n$ have uniform distribution $U(0,1)$.

I can do if there is only $x_1$ and $x_2$ but what about for $n$ number of RVs? Any guideline?

Answer 1

Expand the square. We get $\sum_1^n X_i^2 +2\sum_{1\le i\lt j}X_iX_j$.

Now use the linearity of expectation, and independence. We get $nE(X_1^2)+(n)(n-1)E(X_1)E(X_1)$.