Given the RVs $X_1, X_2, \ldots, X_N$, with each $X_i\sim\mathcal{N}(0,1)$ and $\operatorname{Cov}(X_i,X_j) = 1$.
I want to compute the variance of the sample mean $\overline{X}$ using the general definition of the variance of the sum of RVs:
\begin{align} \operatorname{Var}(\overline{X}) &= \operatorname{Var}\left(\frac1N \sum_{i=1}^{N} X_i\right) \\&= \frac1N \left( \sum_{i=1}^{N} \operatorname{Var}(X_i) + \sum_{i\neq j} \operatorname{Cov}(X_i,X_j)\right) \\&= \frac1N \left(N + \sum_{i\neq j} \operatorname{Cov}(X_i,X_j)\right) \end{align}
Is my approach correct? What is the value of the sum of the covariances?
The factor outside is $\frac 1 { N^{2}}$ and not $\frac 1 N$. The last sum is nothing but $N(N-1)$, the number of choices for $(i,j)$ with $ i \neq j$.