# Ponderate two gaussian mixture

I have 2 independent random variables $$X$$ and $$Y$$ with gaussian mixture distribution like:

$$f(x) = \sum_{i=1}^{m} \phi_{X,i} \mathcal{N}(\mu_{X,i} , \sigma_{X,i}^{2})$$ $$f(y) = \sum_{i=1}^{m} \phi_{Y,i} \mathcal{N}(\mu_{Y,i} , \sigma_{Y,i}^{2})$$

With $$\sum_{i=1}^{m}\phi_{X,i} =\sum_{i=1}^{m}\phi_{Y,i} =1$$.

If I want to ponderate these 2 random variables like: $$Z=\alpha\cdot X + (1-\alpha)Y$$ with $$\alpha\in (0,1)$$, I could say that $$f(z)=\sum_{i=1}^{m} \phi_{Z,i} \mathcal{N}(\mu_{Z,i} , \sigma_{Z,i}^{2})$$?? If I could, so how I can find a relation between the parameters like:

$$\phi_{Z,i} = \phi_{Z,i}(\alpha,\phi_{X,i},\phi_{Y,i})$$ $$\mu_{Z,i} = \mu_{Z,i}(\alpha,\mu_{X,i},\mu_{Y,i})$$ $$\sigma_{Z,i}^{2} = \sigma_{Z,i}^{2}(\alpha,\sigma_{X,i}^{2},\sigma_{Y,i}^{2})$$

for $$i=1,...,m$$. If I have another gaussian mixture $$W$$, I could extend this with $$Z=\alpha_1 X + \alpha_2 Y + \alpha_3 W$$ and $$\sum_{j} \alpha_j =1$$, $$\alpha_j \in (0,1)$$??

• It looks like there is a typo in the first equation: $\sigma_{Y,i} \rightarrow \sigma_{X,i}$. Dec 18 '19 at 15:08
• Thanks, I already corrected it Dec 18 '19 at 15:11

Let $$X_i \sim \mathcal{N}(\mu_{X,i}, \sigma_{X,i})$$ and $$Y_j \sim \mathcal{N}(\mu_{Y,j}, \sigma_{Y,j})$$, where $$i=1...m$$ and $$j=1...n$$. Then variables $$X$$ and $$Y$$ can be expressed as $$$$X = \sum_{i=1}^m \phi_{X,i} X_i, Y = \sum_{j=1}^n \phi_{Y,j} Y_j,$$$$ whereas their sum is $$$$Z = \alpha X + (1-\alpha)Y = \sum_{i=1}^m \alpha\phi_{X,i} X_i + \sum_{j=1}^n (1-\alpha)\phi_{Y,j} Y_j,$$$$ and is distributed as a mixture of $$m+n$$ normal distributions: $$$$f(z) = \sum_{i=1}^m \alpha\phi_{X,i} \mathcal{N}(\mu_{X,i}, \sigma_{X,i}) + \sum_{j=1}^n (1-\alpha)\phi_{Y,j} \mathcal{N}(\mu_{Y,j}, \sigma_{Y,j}) = \sum_{k=1}^{m+n} \phi_{Z,k} \mathcal{N}(\mu_{Z,k}, \sigma_{Z,k}),$$$$ with the obvious re-definitions of the coefficients and the parameters.