Suppose $X_1\cdots X_n$ are correlated normal random variables... such that $E(X_i) = 0$, $Cov(X_i, X_j ) = ρσ^2$ for $i$ not equal to $j$, and $Var(X_i) = σ^2$.
In the case where $ρ = 0.2, σ^2 = 2$ and $n = 100$, find the covariance matrix of the multivariate Normal distribution for $X = (X_1, . . . , X_n)'$ and hence find the variance of $\sum_{i=1}^n X_i$.
All I know is that I apparently need to consider $Y = BX =\sum_{i=1}^n X_i$.
This is the only hint our lecturer gave us. But I am still unsure how to go about doing this.
Thank you
The covariance matrix $Cov(X)$ has $\sigma^2$ on the diagonals and $\rho\sigma^2$ on the off diagonals.
You can express the sum $\sum_{i=1}^n X_i$ as $BX$ where $B$ is the column matrix of all ones. Then using the formula for covariance of $BX$ we get $$ Var(\sum X) = Cov(BX)= B'Cov(X)B. $$
The question is presumably asking you to make these matrices and do the multiplications in Rstudio.