I'm trying to develop some intuition for the fact that the family of Gaussian distributions is closed under addition.
I.e. if $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$, then $Y = \sum_iX_i$ is also normal with $Y \sim \mathcal{N}(\sum_i \mu_i, \sum_i \sigma_i^2)$.
If I graph the PDF of $Y$ for some arbitrary $X_1$ and $X_2$, I just cannot believe that the resulting distribution is Normal - it clearly seems like this distribution could be bi-modal;
Wikipedia has several analytic proofs for this problem, but they are a bit dense and I was hoping to develop some visual intuition for this property. Am I misunderstanding something regarding the graph of the PDF of $Y$?