Gaussian Probability

Cumulative Gaussian probability distribution

We already have proven this theorem; however, we did not go far enough. Hence, we have to do it all over again.

Theorem: The normalized Gaussian probability density distribution

(1 / sqrt(2 pi)) exp(- x^2 / 2)

integrates, over its domain R = (-oo, oo), to one.

Proof: Let I be the required integral

I = (1 / sqrt(2 pi)) integral, from -oo to oo, of exp(- x^2 / 2) dx.

Since x is the dummy variable of integration, we may write this integral in terms of any other dummy variable, say, y, as

I = (1 / sqrt(2 pi)) integral, from -oo to oo, of exp(- y^2 / 2) dy.

Multiply them together, to obtain

I^2 = (1 / (2 pi)) integral, from -oo to oo, of the integral from -oo to oo, of exp(- (x^2 + y^2) / 2)) dx dy.

Change from the rectangular coordinates (x, y) to the polar coordinates (r, theta), employing the 1:1 probability-measure-preserving transformation

r = sqrt(x^2 + y^2) r in ;0, oo)
theta = atan(y / x) theta in [0, 2 pi)
x = r cos(theta) x in R = (-oo, oo)
y = r sin(theta). y in R = (-oo, oo).
dx dy = r d-theta dr
d-theta dr = (1 / sqrt(x^2 + y^2)) dx dy.

Then the integral becomes

I^2 = (1 / (2 pi)) integral, from 0 to oo, of the integral, from 0 to 2 pi, of r exp(- r^2 / 2) d-theta dr.

Perform the 1:1 probability-measure-preserving transformation

u = r^2 / 2 u in [0, oo)
r = sqrt(2 u) r in ;0, oo)
du = r dr,
dr = (1 / sqrt(2 u)) du

to obtain

I^2 = (1 / (2 pi)) integral, from 0 to oo, of the integral, from 0 to 2 pi, of exp(- u) d-theta du.

Now, we come to the new part. Perform the additional 1:1 probability-measure-preserving transformation

s = theta / (2 pi) s in [0, 1)
t = 1 - exp(- u) t in [0, 1)
theta = 2 pi s theta in [0, 2 pi)
u = - ln(1 - t) u in [0, oo)
ds = d-theta / (2 pi)
dt = exp(- u) du,
d-theta = 2 pi ds
du = (1 / (1 - t)) dt

to obtain

I^2 = integral, from 0 to 1, of the integral, from 0 to 1, of ds dt = 1.

Since I obviously is positive, its value is one. QED.

Now, by following the steps backwards, we have obtained the

Corollary: The uniform probability distribution on the square [0, 1)^2 is mapped by the 1:1 probability-measure-preserving transformation

x = sqrt(- 2 ln(1 - t)) cos(2 pi s)
y = sqrt(- 2 ln(1 - t)) sin(2 pi s)

onto the normalized Gaussian probability distribution on the Cartesian plane R^2, where R = (-oo, oo).

We also showed that the general Gaussian probability density function is

(2 pi)^(- n / 2) sqrt(det(A)) exp((- 1 / 2) (x - xo) (1 / A) (x - xo)tr),

where A is a positive-definite symmetric n-by-n matrix, x is a variable horizontal n-dimensional vector, and xo is a constant horizontal n-dimensional vector. Likewise, we showed that, over its domain R^n, its integral is one. With effort, these steps also may be reversed, to provide the 1:1 probability-measure-preserving transformation onto the general Gaussian probability distribution on R^n. However, we do not provide the lengthy details; because, as will be shown next, this procedure turned out to be a blind alley.

Random number generator

There are two families of generators, which purport to provide a random stochastic variable: the analogue thermal noise generator, followed by an A-D (= analogue to digital) converter; or a digital computer program, which is a pseudo-random number generator. Each has its peculiarities and short-comings. Neither provides a truly random number sequence. Donald Knuth describes an algorithm for what is considered to be the best pseudo-random number generator. It provides a uniform distribution of integers on a semi-open interval [0, m), where m is some natural number. Then, division by m yields a uniform distribution of rational numbers on [0, 1). Its predictability is both an advantage and a failing. Its periodicity -- the modulus of periodicity is m -- is another failing. However, iff (= if and only if) the sequence is carried through exactly one period, it is shown by Knuth that the values do not repeat; thus, that each of the values is visited. Therefore, the distribution is uniform. But, it is not exactly uniform for any shorter subsequence. While good, it is not perfect on any of the multitude of other tests of randomness. As a result, a generation of a Gaussian probability distribution does not yield the desired result. The resulting distribution departs from its goal so much as to be not usable.

Construction of the general Gaussian probability distribution

Since a direct generation of the general Gaussian probability distribution yields an unusable result, we may as well not attempt this difficult process. Instead, we have devised the following procedure.

Employ a Knuth pseudo-random generator to obtain a stochastic, but, unordered, set, with a cardinality being a real-number multiple of n, of integers, in the interval [0, 1)..
Normalize their domain onto the interval [0, 1).
Employ the transformation in the foregoing corollary, to map pairs of numbers onto the domain R^2.
Take each contiguous sub-set of n numbers as the coordinates in the n-dimensional space, to map onto the domain R^n. It will be approximately a normalized Gaussian probability distribution.
Compute its first moment -- the mean.
Perform a translation, to place the mean at the origin.
Employ the Jacobi algorithm to find the set of eigen-pairs (eigen value and eigen vector).
Employing these eigen-vectors, rotate to place the principal axes along the coordinate axes.
Divide by the eigen-values, to obtain a spherically-symmetrical distribution; that is, a normalized Gaussian probability distribution.
Multiply by the desired eigen-values; that is, by the eigen-values of the desired positive-definite symmetric moment-matrix A.
Rotate by the desired eigen-vectors of this same matrix A.
Translate by the desired mean vector xo.

Now, we have constructed a set which has exactly the desired Gaussian probability distribution, with the moment-matrix A and the mean-vector xo.