Combining Many Factors: The Gaussian Distribution

Abstract

(a) A common distribution in diverse circumstances; (b) Physical origin of Gaussian distributions; (c) The Gaussian as the limiting form of the binomial; (d) Properties of the Gaussian; (e) How random variables combine; (f) The Gaussian form as a fixed point for convolutions; (g) Multi-variate Gaussians; (h) Gaussians in disguise: particle velocities and stellar masses; (i) Error distributions.

Figure 5.1 shows three diverse examples of naturally occurring distributions—cluster star velocities, human heights, and the distribution of the sum of the last four digits of phone numbers in my address book. They all show a very similar shape—symmetrical, with a “bell curve” shape. They can all be reasonably well fitted by the same mathematical expression, which we will come to shortly. This mathematical form is named the “Gaussian Distribution”, after Carl Friedrich Gauss, but it is often referred to as the “Normal Distribution”, because it occurs so frequently in Nature. An important Physics example is the distribution of molecular velocities. Although the Maxwell–Boltzmann distribution for molecular velocities looks somewhat different, it is really a Gaussian in disguise, as we will see later in this chapter. How can such completely different physical circumstances produce the same result? The answer lies in what happens mathematically when many different factors combine. Before we start to examine how this works, lets take a closer look at the distributions in Fig. 5.1.