The basic Gaussian function is simply:

##y=e^(-x^2)##

where the normal distribution is a specific parameterization:

##f(x | mu, sigma^2) = 1/sqrt(2 pi sigma^2) e^(-(x-mu)^2 /(2 sigma^2))##

The basic Gaussian function is simply:

##y=e^(-x^2)##

We can parameterize is with some additional constants:

##y = A e^(-b(x-c)^2)##

If we want to use it for statistical purposes, we would want to make it into where:

##c## becomes the mean i.e. ##c implies bar x## ##b## becomes the reciprocal of half of the variance, i.e. ##b implies 1/(2sigma^2)## and we choose ##A## such that the integral of the function over all ##x## is ##1##, i.e. ##A implies 1/sqrt(2 pi sigma^2)##

The normal distribution function is then given by

##f(x | mu, sigma^2) = 1/sqrt(2 pi sigma^2) e^(-(x-mu)^2 /(2 sigma^2))##