Illustration of the central limit theorem

This article gives two concrete illustrations of the central limit theorem. Both involve the sum of independent and identically-distributed random variables and show how the probability distribution of the sum approaches the normal distribution as the number of terms in the sum increases. This article gives two concrete illustrations of the central limit theorem. Both involve the sum of independent and identically-distributed random variables and show how the probability distribution of the sum approaches the normal distribution as the number of terms in the sum increases. The first illustration involves a continuous probability distribution, for which the random variables have a probability density function. The second illustration, for which most of the computation can be done by hand, involves a discrete probability distribution, which is characterized by a probability mass function. A free full-featured interactive simulation that allows the user to set up various distributions and adjust the sampling parameters is available through the External links section at the bottom of this page. The density of the sum of two independent real-valued random variables equals the convolution of the density functions of the original variables. Thus, the density of the sum of m+n terms of a sequence of independent identically distributed variables equals the convolution of the densities of the sums of m terms and of n term. In particular, the density of the sum of n+1 terms equals the convolution of the density of the sum of n terms with the original density (the 'sum' of 1 term). A probability density function is shown in the first figure below. Then the densities of the sums of two, three, and four independent identically distributed variables, each having the original density, are shown in the following figures.If the original density is a piecewise polynomial, as it is in the example, then so are the sum densities, of increasingly higher degree. Although the original density is far from normal, the density of the sum of just a few variables with that density is much smoother and has some of the qualitative features of the normal density. The convolutions were computed via the discrete Fourier transform. A list of values y = f(x0 + k Δx) was constructed, where f is the original density function, and Δx is approximately equal to 0.002, and k is equal to 0 through 1000. The discrete Fourier transform Y of y was computed. Then the convolution of f with itself is proportional to the inverse discrete Fourier transform of the pointwise product of Y with itself.