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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: $$\mathcal{N}(\mu_i,\sigma_u^2)\times\Sigma_{i=1}^N\mathcal{N}(\mu_i,\sigma_i^2)$$

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