# Sum of Gaussian mixture and Gaussian scale mixture

What will be the distribution of the sum of two independent random variables, say $X$ and $Y$, when $X$ has a Gaussian mixture distribution (when we take Gaussian distribution with different location parameters), and $Y$ is a Gaussian scale mixture (when we take Gaussian distributions with different scale parameters)?

Let $Z=X+Y$ then, the mixture distribution of $X$ has density $\sum_{i=1}^n \pi_i \mathcal{N}(x;\mu_i,\sigma^2_X)$ and the mixture distribution of $Y$ has density $\sum_{j=1}^m \phi_j \mathcal{N}(x;\mu_Y,\sigma^2_j)$, the symbols $\mathcal{N}(x, \dots)$ being normal densities. (One could as well let both parameters vary in both cases, $X$ and $Y$).

Let $X$ has the mixture distribution with density $f(x)=\sum \pi_i f_i(x)$ and $Y$ the mixture distribition with density $g(x) = \sum \phi_i g_i(x)$, and suppose $X$ and $Y$ are independent. A useful tool for analyzing distribution of sums of independent random variables is the moment generating function (look it upon wikipedia if you didn't see it yet). That is given by $\DeclareMathOperator{\E}{E} M_X(t) = \E e^{tX}$. Then the moment generating function of $X+Y$ is the product $M_X(t) M_Y(t)$. Let the moment generating functions for $X$ mixture component $i$ be $M_i$, for $Y$ mixture component $j$ be $G_j$. Then, by linearity of the expectation operator, we have $$M_X(t) = \E e^{tX} = \int e^{tx} f(x)\; dx \\ = \int e^{tx} \sum_i \pi_i f_i(x)\; dx \\ = \sum_i \pi_i M_i(t)$$ and likewise for $Y$ $G_Y(t) = \sum_j G_j(t)$ and then the moment generating function for $X+Y$ is $$M_X(t)G_Y(t)=\sum_i \pi_i M_i(t) \cdot \sum_j \phi_j G_j(t) \\ = \sum_i \sum_j \pi_i \phi_j M_i(t) G_j(t)$$ so the distribution of the sum is a new mixture distribution with $nm$ component (where $X$ has $n$ components, $Y$ has $m$ components), the weights are the products of the old weights $\pi_i \phi_j$, and the distribution of the component $(ij)$ is the distribution of the sum of independent random variables $X_i+Y_j$.

In your case all of those will be normal distributions, but I leave for you to work out the details. Observe that only the very last step of the argument (that left for you) pends on thse being a normal mixture. So what we have shown is a quite gneral result about the istribution of sums of random variables, each one with a mixture distribution.

• is this also true if both $X$ and $Y$ have a mixture of gaussian distribution? and also where each gaussian in the mixture of both $X$ and $Y$ can have different mean and std? Commented Aug 16, 2023 at 16:22
• @David: Yes ... Commented Aug 19, 2023 at 0:47

If I understand your notation you are asking about the distribution of $Y$ when $Y = X + Z$ where $X \sim \mathcal N(a,b)$, $Z \sim \mathcal N(c,d)$.

In this case $Y$ is Gaussian if $Z$ and $X$ are independent or jointly normal, with mean $a+b$ and a covariance matrix dependent on the relationship between $X$ and $Z$.

Using the word mixture to describe the above arrangement will lead to confusion.

• The term mixture is important as X and Z are not Gaussian. X and Z belong to Gaussian mixture and Gaussian scale mixture distribution. We can say X is weighted sum of Gaussian density with different location parameter as $X=\sum_{i=1}^N w_i N(\mu_i,\Sigma)$ and Z is weighted sum of Gaussian with different scale parameter. $Z=dU$ where $U \sim \mathdd{N}(0,Q)$ and $d$ is scalar random variable with density as mixing distribution. Commented Sep 30, 2015 at 12:47
• Try being more clear with your descriptions. You said above "X is just a sum of Gaussians." That is not a mixture. You've picked a bad choice of notation if you wish $X$, $Y$ and $Z$ to represent distribution functions rather than random variables and want people to implicitly guess that. Commented Sep 30, 2015 at 13:19
• Writing $X=\sum_{i=1}^N w_i N(\mu_i,\Sigma)$ perpetuates the confusion. Do you mean $X\sim\sum_{i=1}^N w_i N(\mu_i,\Sigma)$ as in a mixture model? Or $X=\sum_{i=1}^N w_i X_i$ with $X_i\sim N(\mu,\Sigma)$ as in a Gaussian weighted average? Commented Sep 30, 2015 at 13:50
• This is impenetrable and I thus vote to close this question as "unclear". Commented Oct 1, 2015 at 11:31
• May I know what is unclear , so that I can edit my question accordingly. I just want distribution where X belongs to gaussian mixture model and Y belong to Gaussian scale mixture. Commented Oct 2, 2015 at 9:13