I am about to compute the variance of the following two variables: $u_n=2.\frac{X_1+X_2+ \ldots +X_n}{n}-1$ and $w_n=max\{X_1,X_2, \ldots ,X_n\}$ which almost surely converge to $m$ I evaluated before.
I have done $E(u_n)=m$ and $E(w_n)=m-\frac{1^n+\ldots+(m-1)^n}{m^n}$
How can I evaluate the variance? through the expectation or $D(u_n)=D(\frac{X_1+X_2+ \ldots +X_n}{n}-1)$ (how to eliminate the brackets?)? I have no idea with that especially $w_n$. Hope any help!
$D(u_n)$ is the variance of $u_n$ i mean.
$(X_i)_{i=1,\ldots,n}$ is an i.i.d sequence that has distribution $1/m$ obtaining value $1,2,\ldots \quad or \quad m$ with probability $1/m$.
Am I right that: $$D(u_n)=D(\frac{X_1+X_2+ \ldots +X_n}{n}-1)= \frac{1}{n^2}D(X_1)=\frac{1}{n^2}[E(X_1^2)-(E(X_1))^2]$$ then apply that $$E(X_1^2)=\frac{1}{m}(1^2+\ldots+m^2)$$