# What is the PDF of a division/product of a Gaussian PDF and a mixed Gaussian PDF

Let we use $$\mathcal{N}(\mu,\sigma^2)$$ denote a gaussian distribution, and $$\Sigma_{i=1}^N\mathcal{N}(\mu_i,\sigma_i^2)$$ is a mixed gaussian distribution.

What is the distribution of their division: $$\frac{\mathcal{N}(\mu_i,\sigma_u^2)}{\Sigma_{i=1}^N\mathcal{N}(\mu_i,\sigma_i^2)}$$

Moreover, what is the distribution of their product: $$\mathcal{N}(\mu_i,\sigma_u^2)\times\Sigma_{i=1}^N\mathcal{N}(\mu_i,\sigma_i^2)$$