# Undestanding Scaling factor of a normal distribution and product of two normal distributions.

Product of two normal distribution is only proportional to a normal distribution. Therefore in order to have a normal distribution it has to be scaled.

# Scale factor for Product of two gaussians

## Gaussian function $yy_1$ - with $\mu_1$ = 0 and $\sigma_1$ = 1

\begin{align} yy_1 = \frac{1}{\sqrt{2\pi\sigma_1^2}}e^{-\frac{(x-\mu_1)^2}{2\sigma_1^2}} \end{align}

\begin{align} \mu_1 = 0, \sigma_1 = 1 \end{align}

## Gaussian function $yy_2$ - with $\mu_2$ = 0 and $\sigma_2$ = 5

\begin{align} yy_2 = \frac{1}{\sqrt{2\pi\sigma_2^2}}e^{-\frac{(x-\mu_2)^2}{2\sigma_2^2}} \end{align}

\begin{align} \mu_2 = 0, \sigma_2 = 5 \end{align}

## Area under the curve $yy_1$ and $yy_2$

(0.9990000000000004, 0.9989367151430915)

## Product of $yy_1$ and $yy_2$ and its plot

\begin{align} yy_1 * yy_2 = \frac{1}{2\pi\sigma_1^2\sigma_2^2}e^{\frac{(x - \mu_1)^2}{2\sigma_1^2}}e^{\frac{(x - \mu_2)^2}{2\sigma_2^2}} \end{align}

## Area under the curve for the gaussian product

0.07816077915736717

# The correct scaling factor

0.07823901817554269

'Area under the curve : 0.9990000000000001'