# What is the distribution of the weighted sum of two multivariate normal random variables?

Say we have two multivariate normal random variables $$X$$ and $$Y$$ with the same dimensionality, means $$\mu_x$$ and $$\mu_y$$, and covariance matrices $$\Sigma_x$$ and $$\Sigma_y$$. It is possible to show that the distribution of $$X+Y$$ is also multivariate random normal with mean $$\mu_x + \mu_y$$ and covariance $$\Sigma_x+\Sigma_y$$, but what if the distributions are not equally weighted?

Formally, what is the distribution of the sum of $$p_xX+(1-p_x)Y$$ where $$p_x\in(0,1)$$ is the proportion of the sum attributable to the distribution $$X$$?

• What's the distribution of $p_xX$? Aug 4, 2021 at 23:13
• Are $X$ and $Y$ independent of each other? Aug 5, 2021 at 0:56

Any linear combination of independent multivariate normals of the same dimension is also multivariate normal. Therefore, if we assume $$X$$ and $$Y$$ are independent then
$$Z_p = pX + (1-p)Y \sim \mathcal{N}\left(p\mu_X + (1-p)\mu_Y, p^2\Sigma_X+ (1-p)^2\Sigma_Y\right)$$