The sum of Two Identical Normals is Normal: Difference between revisions

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Normal Density function

Normally distributed variable x with mean ${\displaystyle \left.\mu \right.}$ and standard deviation ${\displaystyle \left.\sigma \right.}$ has density function

${\displaystyle d(x)={\frac {1}{{\sqrt {2\pi }}\sigma }}e^{\frac {-(x-\mu )^{2}}{2\sigma }}}$

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:

 ${\displaystyle \left.d(s)\right.}$ ${\displaystyle =\int _{-\infty }^{\infty }d_{1}(x)d_{2}(s-x)dx}$ ${\displaystyle ={\frac {1}{\left({\sqrt {2\pi }}\sigma \right)^{2}}}\int _{-\infty }^{\infty }e^{\frac {-(x-\mu )^{2}}{2\sigma ^{2}}}e^{\frac {-(s-x-\mu )^{2}}{2\sigma ^{2}}}dx}$ ${\displaystyle ={\frac {1}{\left({\sqrt {2\pi }}\sigma \right)^{2}}}\int _{-\infty }^{\infty }e^{\frac {-(x-\mu )^{2}-(s-x-\mu )^{2}}{2\sigma ^{2}}}dx}$ ${\displaystyle ={\frac {1}{\left({\sqrt {2\pi }}\sigma \right)^{2}}}\int _{-\infty }^{\infty }e^{\frac {2(x^{2}+\mu ^{2})-2(x-\mu )s-s^{2}}{-2\sigma ^{2}}}dx}$ ${\displaystyle ={\frac {1}{\left({\sqrt {2\pi }}\sigma \right)^{2}}}\int _{-\infty }^{\infty }e^{\frac {2(x-s/2)^{2}+(s-2\mu )^{2}/2}{-2\sigma ^{2}}}dx}$ ${\displaystyle ={\frac {1}{\left({\sqrt {2\pi }}\sigma \right)^{2}}}\int _{-\infty }^{\infty }e^{\frac {-2(x-s/2)^{2}}{2\sigma ^{2}}}e^{\frac {-(s-2\mu )^{2}/2}{2\sigma ^{2}}}dx}$ ${\displaystyle ={\frac {1}{{\sqrt {2\pi }}\left({\frac {\sigma }{\sqrt {2}}}\right)}}\int _{-\infty }^{\infty }e^{\frac {-(x-s/2)^{2}}{2\left({\frac {\sigma }{\sqrt {2}}}\right)^{2}}}dx\left({\frac {1}{{\sqrt {2\pi }}\left({\sqrt {2}}\sigma \right)}}e^{\frac {-(s-2\mu )^{2}}{2\left({\sqrt {2}}\sigma \right)^{2}}}\right)}$ ${\displaystyle ={\frac {1}{{\sqrt {2\pi }}\left({\sqrt {2}}\sigma \right)}}e^{\frac {-(s-2\mu )^{2}}{2\left({\sqrt {2}}\sigma \right)^{2}}}}$

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

More Generally

More generally, the sum of two normals is normal, with parameters mean

${\displaystyle \left.\mu _{X+Y}=\mu _{X}+\mu _{Y}\right.}$

and variance

${\displaystyle \sigma _{X+Y}^{2}=\sigma _{X}^{2}+\sigma _{Y}^{2}}$

By induction, the sum of n normals will be normal, with parameters

${\displaystyle \mu _{\sum _{i=1}^{n}X_{i}}=\sum _{i=1}^{n}\mu _{X_{i}}}$

and

${\displaystyle \sigma _{\sum _{i=1}^{n}X_{i}}^{2}=\sum _{i=1}^{n}\sigma _{X_{i}}^{2}}$