Cauchy distribution--the normal distribution of the brothers

One of the three most important features of the variable distribution is fat-thin (the other two: Singlemode-multimode; symmetry-biased), Cauchy distribution and normal distribution are very confusing distribution curves.

Cauchy distribution is also called Cauchy-Lorentz distribution , which is a continuous probability distribution named after Augustin Louis Cauchy and Hendrick Lorenze, and its probability density function is

where x0 is the positional parameter that defines the peak position of the distribution,Gamma is the scale parameter of half the width of the maximum value.

As a probability distribution, often called Cauchy distribution , physicists also refer to it as Lorentz distribution or breit-wigner distribution . A large part of the importance of physics is attributed to the solution of a differential equation that describes forced resonance. In spectroscopy, it describes the shape of a spectral line that is widened by resonance or other mechanisms. The following section will use Cauchy to distribute this statistical term.

The special case of x0 = 0 and γ = 1 is called the standard Cauchy distribution , and its probability density function is

Characteristics

Its cumulative distribution function is:

The inverse cumulative distribution function of Cauchy distribution is

The mean, variance, or moment of the Cauchy distribution is undefined, and its majority and median values are defined equal to x0.

The X represents the Cauchy distributed random variable, and the Cauchy distribution's characteristic function is expressed as:

If you and v are two independent normal distribution random variables with expected value of 0 and variance of 1, then the ratio u/v is Cauchy distributed.

The standard Cauchy distribution is a special case where the student's T-distribution degree of freedom is 1.

The Cauchy distribution is a stable distribution: if, then.

If x1, ..., xn are mutually independent random variables that conform to Cauchy distribution, then the arithmetic averages (x1 + ... + xn)/n have the same Cauchy distribution. To prove this, let's calculate the characteristic function of the sampling average:

Among them, is the sampling average. This example shows that the finite variable hypothesis in the central limit theorem cannot be discarded.

Lorentz linear distribution is more suitable for the relatively flat, wide curve Gaussian linear distribution is suitable for a higher, narrower curve of course, if it is a more centered situation, both can. In many cases, the use of both is a percentage of the practice. such as Lorenz accounted for 60%, Gauss accounted for 40%.

Cauchy-Lorentz distribution

Probability density function The Green Line is the standard Cauchy distribution
Cumulative distribution function Corresponds to the color in
Parameters	Positional parameters (real numbers) Scale parameter (real number)
Range
Probability density function
Cumulative distribution function
Mark	{{{notation}}}
Expectations	(Not defined)
Number of Median
The majority of
Variance	(Not defined)
Partial state	(Not defined)
Peak State	(Not defined)
Entropy value
The function of the moving and inferior	(Not defined)
Feature functions

External links

• Eric Westini, Cauchy distribution at MathWorld
• GNU Scientific Library-reference Manual

Cauchy distribution--the normal distribution of the brothers