Notes on Gaussian functions, the Gaussian integral, and the Normal Distribution

The Normal distribution is a Gaussian probability distribution. Gaussian probability distributions are functions designed to reflect principles of the central limit theorem which states that a population sample will tend towards the expected value with a sufficiently large random sample and that values farther away from the expected value will occur less frequently. Then there’s the Gaussian integral which is the definite integral of a Gaussian function integrated over the entire real line. These three topics, Gaussian functions, the Gaussian Integral, and Gaussian probability distributions are so inter-woven that I thought it would be best to attempt to address all three of them in one go (I was wrong about that but that’s a topic for a different article). First, we’ll look into what the general definition of a Gaussian function is, then we’ll take a look at the Gaussian integral whose result is necessary in determining the normalization constant for the normal distribution. Lastly, we’ll use the information and understanding we’ve collected and derive the equation for the normal distribution.

First, lets get a clear understanding of what a Gaussian function actually is. A Gaussian function (or just “Gaussian”) is a function that composes the exponential function exp(x) with a concave quadratic function such as −(ax^2+bx+c) or −(ax^2+bx) or just −ax^2. The result is a family of functions that take on the shape of the infamous “bell curve”.

A graph of two Gaussians. The first Gaussian (green) has λ=1 and a=1. The second (orange) has λ=2 and a=1.5. Neither function is normalized. That is, the area under these curves does not equal 1.

Most people are familiar with these classes of curves because they’re so prolifically used in probability and statistics most notably as the probability density function of a normally distributed random variable. In these cases the function has coefficients and parameters that both scale the amplitude of the “bell”, vary its standard deviation (its width), and translate the mean, all while normalizing the area under the curve (scaling the bell so that the area under the curve always equals 1)…