Illustration of The Continuous Case
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.
Read more about this topic: Illustration Of The Central Limit Theorem
Famous quotes containing the words illustration of, illustration, continuous and/or case:
“Each truth that a writer acquires is a lantern, which he turns full on what facts and thoughts lay already in his mind, and behold, all the mats and rubbish which had littered his garret become precious. Every trivial fact in his private biography becomes an illustration of this new principle, revisits the day, and delights all men by its piquancy and new charm.”
—Ralph Waldo Emerson (1803–1882)
“An illustration is no argument,—nor do I maintain the wiping of a looking-glass clean, to be a syllogism;Mbut you all, may it please your worships, see the better for it.”
—Laurence Sterne (1713–1768)
“Perhaps when distant people on other planets pick up some wave-length of ours all they hear is a continuous scream.”
—Iris Murdoch (b. 1919)
“In the case of a deliberate, palpable, and dangerous exercise of ... powers not granted by the compact, the States ... are in duty bound to interpose for arresting the progress of the evil, and for maintaining within their respective limits the authorities, rights, and liberties appertaining to them.”
—James Madison (1751–1836)