# The sum of Two Identical Normals is Normal

## Normal Density function

Normally distributed variable x with mean $\left.\mu\right.$ and standard deviation $\left.\sigma\right.$ has density function $d(x)=\frac{1}{\sqrt{2\pi}\sigma}e^{\frac{-(x-\mu)^2}{2\sigma}}$

## The Distribution of the Sum of Two iid Normal Variables

Consider the sum s of two of these random variables x. The density of s is given by the convolution of the densities of the two: $\left.d(s)\right.$ $=\int_{-\infty}^{\infty}d_1(x)d_2(s-x)dx$ $=\frac{1}{\left(\sqrt{2\pi}\sigma\right)^2}\int_{-\infty}^{\infty}e^{\frac{-(x-\mu)^2}{2\sigma^2}}e^{\frac{-(s-x-\mu)^2}{2\sigma^2}}dx$ $=\frac{1}{\left(\sqrt{2\pi}\sigma\right)^2}\int_{-\infty}^{\infty}e^{\frac{-(x-\mu)^2-(s-x-\mu)^2}{2\sigma^2}}dx$ $=\frac{1}{\left(\sqrt{2\pi}\sigma\right)^2}\int_{-\infty}^{\infty}e^{\frac{2(x^2+\mu^2)-2(x-\mu)s-s^2}{-2\sigma^2}}dx$ $=\frac{1}{\left(\sqrt{2\pi}\sigma\right)^2}\int_{-\infty}^{\infty}e^{\frac{2(x-s/2)^2+(s-2\mu)^2/2}{-2\sigma^2}}dx$ $=\frac{1}{\left(\sqrt{2\pi}\sigma\right)^2}\int_{-\infty}^{\infty}e^{\frac{-2(x-s/2)^2}{2\sigma^2}}e^{\frac{-(s-2\mu)^2/2}{2\sigma^2}}dx$ $=\frac{1}{\sqrt{2\pi}\left(\frac{\sigma}{\sqrt{2}}\right)} \int_{-\infty}^{\infty}e^{\frac{-(x-s/2)^2}{2\left(\frac{\sigma}{\sqrt{2}}\right)^2}}dx \left( \frac{1}{\sqrt{2\pi}\left(\sqrt{2}\sigma\right)} e^{\frac{-(s-2\mu)^2}{2\left(\sqrt{2}\sigma\right)^2}} \right)$ $= \frac{1}{\sqrt{2\pi}\left(\sqrt{2}\sigma\right)} e^{\frac{-(s-2\mu)^2}{2\left(\sqrt{2}\sigma\right)^2}}$

Conclusion: the sum is normally distributed, with mean $\left.2\mu\right.$, and with standard deviation $\sqrt{2}\sigma$.

## More Generally

More generally, the sum of two normals is normal, with parameters mean $\left.\mu_{X+Y}=\mu_{X}+\mu_{Y}\right.$

and variance $\sigma_{X+Y}^2=\sigma_{X}^2+\sigma_{Y}^2$

By induction, the sum of n normals will be normal, with parameters $\mu_{\sum_{i=1}^nX_i}=\sum_{i=1}^n\mu_{X_i}$

and $\sigma_{\sum_{i=1}^nX_i}^2=\sum_{i=1}^n\sigma_{X_i}^2$