# A general expression for the sum of multiple independent Normal Mixture Distributions?

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?

• Isn't the sum the mixture of $2^n$ different Normals, with boring means and variances? – kimchi lover Dec 6 '19 at 0:49
• 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. – user120911 Dec 6 '19 at 0:53

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 $$P(I=i,J=j) = P(I=i)P(J=j) = w_{X,i}w_{Y,j}$$.