Illustration of the central limit theorem |
Here is an illustration of the central limit
theorem. A probability density
function is shown in the first figure. Then the densities of the sums of two, three, and four independent variables, each having the original density, are shown in the later figures. 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.
A more concrete illustration, in which most of the arithmetic can be done more-or-less instantly by hand, is at concrete
illustration of the central limit theorem.
The densities of the sums of two, three, and four terms were constructed as the convolution of the original density with itself. As the original density is a piecewise polynomial (of degree 0 and 1), the convolutions are
also piecewise polynomials, of increasing degree. Thus the convolution of the original density may be considered a means of
constructing a piecewise polynomial approximation to 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.
We start with a probability density
function. This function, although discontinuous, is far from the most pathological example that could be created.
This density has mean 0 and standard deviation 1.
Next we compute the density of the sum of two independent variables, each having the above density. The density of the sum is
the convolution of the above density with itself.
The sum of two variables has mean 0 but it has standard deviation √2. The density shown in the figure at right has been
rescaled so that its standard deviation is 1.
This density is already smoother than the original. There are obvious lumps, which correspond to the intervals on which the
original density was defined.
We then compute the density of the sum of three independent variables, each having the above density. The density of the sum
is the convolution of the first density with the second.
The sum of three variables has mean 0 but it has standard deviation √3. The density shown in the figure at right has
been rescaled so that its standard deviation is 1.
This density is even smoother than the preceding one. The lumps can hardly be detected in this figure.
Finally, we compute the density of the sum of four independent variables, each having the above density. The density of the
sum is the convolution of the first density with the third.
The sum of four variables has mean 0 but it has standard deviation 2 = √4. The density shown in the figure at right has
been rescaled so that its standard deviation is 1.
This density appears qualitatively very similar to a normal density. Any lumps cannot be distinguished by the eye.
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