01. Gauß

A Gauß Distribution is what you get over time when you calculate the sum of random variables and plot the sums over time (in the limit for infinite count).

Gauß Distribution

I:=∫\limits_{-∞}^{∞} \exp[-b⋅(w-w_0)²]⋅dw

We substitute:

y:=w-w_0
dy=dw

And get:

I:=∫\limits_{-∞}^{∞} \exp[-b⋅y²]⋅dy

We calculate by detour into a higher dimension:

I²=(∫\limits_{-∞}^{∞} \exp[-b⋅y²]⋅dy)²
I²=∫\limits_{-∞}^{∞} \exp[-b⋅x²]⋅dx⋅∫\limits_{-∞}^{∞} \exp[-b⋅y²]⋅dy
I²=∫\limits_{-∞}^{∞} ∫\limits_{-∞}^{∞} \exp[-b⋅x²]⋅dx⋅\exp[-b⋅y²]⋅dy
I²=∫\limits_{-∞}^{∞} ∫\limits_{-∞}^{∞} \exp[-b⋅(x²+y²)]⋅dx⋅dy

We use polar coordinates:

x:=r⋅\cos φ
y:=r⋅\sin φ
x²+y²=r²
dx⋅dy=r⋅dr⋅dφ

And get:

I²=∫\limits_{0}^{2⋅π} ∫\limits_{0}^{∞} \exp[-b⋅r²]⋅r⋅dφ⋅dr
I²=2⋅π⋅∫\limits_{0}^{∞} \exp[-b⋅r²]⋅r⋅dr

We substitute:

z:=-b⋅r²
÷{dz}{dr}=-b⋅2⋅r
÷{dz}{-2⋅b⋅r}=dr

And get (we use a generic integral here and substitute into the result):

-÷{π}{b}⋅∫ \exp[z]⋅dz=2⋅π⋅∫ \exp[z]⋅r⋅÷{dz}{-2⋅b⋅r}
-÷{π}{b}⋅∫ \exp[z]⋅dz=-÷{π}{b}⋅∫ \exp[z]⋅dz
-÷{π}{b}⋅∫ \exp[z]⋅dz=-÷{π}{b}⋅\exp[z]+C
-÷{π}{b}⋅∫ \exp[z]⋅dz=-÷{π}{b}⋅\exp[-b⋅r²]+C
I²=-÷{π}{b}⋅(\exp[-b⋅∞]-\exp[-b⋅0])

If b is complex, we split b into a real and imaginary part:

b=g+i⋅h
I²=-÷{π}{b}⋅(\exp[-(g+i⋅h)⋅∞]-\exp[-b⋅0])
I²=-÷{π}{b}⋅(\exp[-(g⋅∞+i⋅h⋅∞)]-\exp[-b⋅0])
I²=-÷{π}{b}⋅(\exp[-g⋅∞]⋅\exp[-i⋅h⋅∞]-\exp[-b⋅0])

We assume that g≥0:

I²=-÷{π}{b}⋅(0⋅\exp[-i⋅h⋅∞]-\exp[-b⋅0])
I²=-÷{π}{b}⋅(0-1)
I²=÷{π}{b}
I=√{÷{π}{b}}

Normal Distribution

The standard form of the Gauß Distribution (called Normal Distribution) is:

f[x]:=÷{1}{√{2⋅π}⋅σ}⋅\exp[-÷{(x-x_0)^2}{2⋅σ^2}]

Variance

σ²=V=2⋅b

Standard Deviation

68.27% of all values have only a deviation σ from the arithmetic mean.

95.45% of all values have only a deviation of 2⋅σ from the arithmetic mean.

99.73% of all values have only a deviation of 3⋅σ from the arithmetic mean.

full width at half maximum (FWHM)

The FWHM is the distance between two function arguments where the function value is half of maximum (the maximum being ÷{1}{√{2⋅π}} here).

FWHM=σ⋅2⋅√{2⋅\ln 2}
FWHM\approx 2.35⋅σ

Author: Danny (remove the ".nospam" to send)

Last modification on: Thu, 09 May 2013 .