Question

What is the Gaussian function?

Answer 1

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))`

Explanation:

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 the standard normal distribution 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 probability distribution function is then given by

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