Variance of sum of dependent Random Variables

Question

Let X be a Gaussian random variable, but all X_1 is not independent of X_2, X_2 of X_3, etc. Let Y = sum of all n X's, what is the variance?

So if n = 2, then Var(Y) = Var(X_1) + Var(X_2) + 2 * Cov(X_1,X_2), but what about 3, 4, ...?

Thanks

Answer 1

In general (not just for Gaussian random variables), we have the following:

$$\text{Var}\left( \sum_{i=1}^n X_i\right)=\sum_{i=1}^n \text{Var}(X_i)+2\sum_{i<j}\text{Cov}(X_i,X_j)$$