What is the Gaussian function?

1 Answer
David B.
Jul 15, 2016

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))#

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