[Math] Sum of two Gaussian PDFs

Question

I want to find the sum of two univariate Gaussians with different variances.
Considering that the sum of two independent normal variables $X_1$ and $X_2$ gives $X_1+X_2 ~ N(\mu_1+\mu_2, \sigma^2_1+\sigma^2_2)$, then I would expect that the sum of their PDFs is another Gaussian PDF. That is $G_{\sigma^2_1+\sigma^2_1} (x)$. Is that the case? What happens if they are independent?

Answer 1

PDFs must integrate 1, so the sum of two PDFs integrates 2 (it cannot be a PDF). Your question is linked to random variable mixtures, where you define a random variable whose PDF is the convex combination of the original PDFs. For example the average of two normals is a mixture of normals (two bells).

I recommend you read: https://en.wikipedia.org/wiki/Mixture_distribution#Two_normal_distributions