# sum of 3 correlated jointly random variables

Suppose we have $$3$$ jointly normally distributed random variables $$X_1, X_2, X_3$$ with mean $$0$$ and variances $$\sigma_1^2, \sigma_2^2, \sigma_3^2$$. Suppose their correlations are $$\rho_{12}, \rho_{23}, \rho_{13}$$.

Now I know that $$X_1 + X_2 \sim N(0, \sigma_1^2 + \sigma_2^2 + 2 \rho_{12} \sigma_1 \sigma_2)$$, but I'm not sure how to extend this to get the distribution of $$X_1 + X_2 + X_3$$. I'm guessing that it is $$X_1 + X_2 + X_3 \sim N(0, \sigma_1^2 + \sigma_2^2 + \sigma_3^2 + 2 \rho_{12} \sigma_1 \sigma_2 + 2 \rho_{13} \sigma_1 \sigma_3 + 2 \rho_{23} \sigma_2 \sigma_3)$$ but I find it hard to actually prove it. I don't really know the correlation between $$X_1 + X_2$$ and $$X_3$$ so I can't just repeat this twice.

How would I do this?

You can explicitly write \begin{align} \operatorname{Var}[X_1+X_2+X_3] &= \mathbb{E}[(X_1+X_2+X_3)^2] = \mathbb{E}[X_1^2+X_2^2+X_3^2 + 2X_1X_2 + 2X_1X_3 + 2X_2X_3] \\ &=\mathbb{E}[X_1^2]+\mathbb{E}[X_2^2]+\mathbb{E}[X_3^2] + 2\mathbb{E}[X_1X_2] + 2\mathbb{E}[X_1X_3] + 2\mathbb{E}[X_2X_3] \\ &= \sigma_1^2+\sigma_2^2+\sigma_3^2 +2\rho_{12}\sigma_1\sigma_2 +2\rho_{13}\sigma_1\sigma_3 +2\rho_{23}\sigma_2\sigma_3 \end{align} as you thought.
• (You can also go there, if you are comfortable with that, by recalling that $\operatorname{Cov}$ is bilinear, so $\operatorname{Cov}(X_1+X_2,X_3)=\operatorname{Cov}(X_1,X_3)+\operatorname{Cov}(X_2,X_3)$) Apr 10, 2019 at 18:40