Correlated Sum Variance

Expand $\text{Var}(X+Y)$ in terms of $\text{Var}(X)$, $\text{Var}(Y)$, and $\text{Cov}(X,Y)$.

Recall $\text{Cov}(X,Y)=\rho \sqrt{\text{Var}(X)\text{Var}(Y)}$.

1. 1. Expand the variance of a sum

   $\text{Var}(X+Y)=\text{Var}(X)+\text{Var}(Y)+2\text{Cov}(X,Y)$.

   $$\text{Var}(X+Y)=\text{Var}(X)+\text{Var}(Y)+2\text{Cov}(X,Y)$$

2. 2. Substitute given quantities

   With $\text{Var}(X)=\text{Var}(Y)=1$ and $\text{Cov}(X,Y)=\rho \cdot 1\cdot 1=\rho$, we get $\text{Var}(X+Y)=1+1+2\rho =2+2\rho$.

   $$\text{Var}(X+Y)=2+2\rho$$

Answer: 2 + 2rho