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Suppose random variable $X_1$ is a mixture of two Normal distributions with means of $\mu_A$ and $\mu_B$ respectively, standard deviations of $\sigma_A$ and $\sigma_B$ respectively, and weights given by $w_{1_A}$ and $w_{1_B}$. Suppose further that another random variable exists, $X_2$ , which is independent of $X_1$, but is also a mixture of two Normal distributions. Furthermore, $X_2$ has the same means and standard deviations as $X_1$, but has its own weights given by $w_{2_A}$ and $w_{2_B}$. Indeed, suppose there are $n$ many independent random variables like $X_1$ and $X_2$, each with their own weights.

Does a general expression exist for the sum of all $X$'s?

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  • $\begingroup$ Isn't the sum the mixture of $2^n$ different Normals, with boring means and variances? $\endgroup$ Dec 6, 2019 at 0:49
  • $\begingroup$ I have been unable to find any mention of such sums, not even the sum of two gaussian mixtures. Any insight would be helpful and appreciated. $\endgroup$
    – user120911
    Dec 6, 2019 at 0:53

1 Answer 1

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I have restated what I am assuming your question to be. Let $f_{\mu, \sigma^2}(x)$ be the density of a Gaussian with mean $\mu$ and variance $\sigma^2$. Consider mixtures $X$ and $Y$ with respective densities $$ f_X(x) = \sum_{i=1}^{n_X} w_{X,i} f_{\mu_{X,i}, \sigma^2_{X,i}}(x),\quad f_Y(y) = \sum_{j=1}^{n_Y} w_{Y, j} f_{\mu_{Y, j}, \sigma^2_{Y, j}}(y),$$ where $\mu_{\cdot, \cdot} \in \mathbb{R}$ and $\sigma^2_{\cdot, \cdot} > 0$. You are assuming $X$ and $Y$ are independent and you want the density of their sum, $Z = X+Y$. In other words, you want the convolution of the densities $f_X(x)$ and $f_Y(y)$. This works out to be $$f_{Z}(z) = \sum_{i=1}^{n_X}\sum_{j=1}^{n_X} w_{X,i}w_{Y,j} f_{\mu_{X,i}+\mu_{Y,j}, \sigma^2_{X,i} + \sigma^2_{Y, j}}(z).$$ To see this, consider the label random variables $I$ and $J$, which are independent and satisfy $$P(I=i) = w_{X,i},\quad P(J=j) = w_{Y, j}.$$ Then you could have simply defined $X$ and $Y$ by $$X = X_I = \sum_{i=1}^{n_X} 1\{I=i\} X_i, \quad Y = Y_J = \sum_{j=1}^{n_Y} 1\{J=j\} Y_j, $$ where $X_i \sim N(\mu_{X,i}, \sigma^2_{X,i})$ and $Y_j \sim N(\mu_{Y,j}, \sigma^2_{Y,j})$. Since $J$, $I$, $X_i$, $Y_j$ are all mutually independent, conditioned on $I=i$ and $J=j$, $$X + Y = X_i + Y_j,$$ which is the sum of two independent Gaussians, and so has density $f_{\mu_{X,i}+\mu_{Y,j}, \sigma^2_{X,i} + \sigma^2_{Y, j}}(z)$. The event $I=i$, $J=j$ occurs with probability $P(I=i,J=j) = P(I=i)P(J=j) = w_{X,i}w_{Y,j}$.

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