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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 $X$?

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

1 Answer 1

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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

$$Z_p = pX + (1-p)Y \sim \mathcal{N}\left(p\mu_X + (1-p)\mu_Y, p^2\Sigma_X+ (1-p)^2\Sigma_Y\right)$$

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