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;

Example graph of the PDF of Y = X_1 + X_2

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 $Y$?

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    $\begingroup$ Density of X+Y is NOT $f_X(x)+f_Y(y)$, which is what you graphed. $\endgroup$ – Mdoc May 15 at 23:24
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    $\begingroup$ @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. $\endgroup$ – Jam May 15 at 23:30
  • $\begingroup$ math.la.asu.edu/~jtaylor/teaching/Fall2010/STP421/lectures/… $\endgroup$ – Shogun May 15 at 23:39
  • $\begingroup$ @Jam Post owners will be notified in case of comments, and there's no need to tag them. $\endgroup$ – GNUSupporter 8964民主女神 地下教會 May 15 at 23:51

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