# Visual Intuition: Gaussians closed under addition

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 $$Y$$?

• Density of X+Y is NOT $f_X(x)+f_Y(y)$, which is what you graphed. – Mdoc May 15 at 23:24
• @aaron Intuitively, you know that the average of $X+Y$ should be around the sum of the averages of $X$ and $Y$ so it doesn't make sense for it to be bimodal. Also intuitively, if $\int_\mathbb{R}f_X=1$ and $\int_\mathbb{R}f_Y=1$ then adding those two PDFs would give you $\int_\mathbb{R}f_X+f_Y=2$, which can't be right. – Jam May 15 at 23:30
• math.la.asu.edu/~jtaylor/teaching/Fall2010/STP421/lectures/… – Shogun May 15 at 23:39
• @Jam Post owners will be notified in case of comments, and there's no need to tag them. – GNUSupporter 8964民主女神 地下教會 May 15 at 23:51