Cauchy distribution

Cauchy-Lorentz

Probability density function The green line is the standard Cauchy distribution
Cumulative distribution function Colors match the pdf above
Parameters	$x_{0}\!$ location (real) $\gamma >0\!$ scale (real)
Support	$x\in (-\infty ;+\infty )\!$
PDF	${\frac {1}{\pi \gamma \,\left[1+\left({\frac {x-x_{0}}{\gamma }}\right)^{2}\right]}}\!$
CDF	${\frac {1}{\pi }}\arctan \left({\frac {x-x_{0}}{\gamma }}\right)+{\frac {1}{2}}$
Mean	not defined
Median	$x_{0}$
Mode	$x_{0}$
Variance	not defined
Skewness	not defined
Excess kurtosis	not defined
Entropy	$\ln(4\,\pi \,\gamma )\!$
MGF	not defined
CF	$\exp(x_{0}\,i\,t-\gamma \,|t|)\!$

The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous probability distribution. As a probability distribution, it is known as the Cauchy distribution while among physicists it is known as a Lorentz distribution, a Lorentz(ian) function or the Breit-Wigner distribution.

Its importance in physics is due to it being the solution to the differential equation describing forced resonance. In spectroscopy it is the description of the line shape of spectral lines which are subject to homogeneous broadening in which all atoms interact in the same way with the frequency range contained in the lineshape. Many mechanisms cause homogeneous broadening, most notably collision broadening.

Characterization

Probability density function

The Cauchy distribution has the probability density function

${\begin{aligned}f(x;x_{0},\gamma )&={\frac {1}{\pi \gamma \left[1+\left({\frac {x-x_{0}}{\gamma }}\right)^{2}\right]}}\\[0.5em]&={1 \over \pi }\left[{\gamma \over (x-x_{0})^{2}+\gamma ^{2}}\right]\end{aligned}}$

where x₀ is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). The amplitude of the above lorentzian function is given by

${\text{Amplitude (or height)}}={\frac {1}{\pi \gamma }}$

In physics three parameter Lorentzian function is often used as follows:

${\begin{aligned}f(x;x_{0},\gamma ,I)&={\frac {I}{\left[1+\left({\frac {x-x_{0}}{\gamma }}\right)^{2}\right]}}\\[0.5em]&={I}\left[{\gamma ^{2} \over (x-x_{0})^{2}+\gamma ^{2}}\right]\end{aligned}}$

where I is the height of the peak. The special case when x₀ = 0 and γ = 1 is called the standard Cauchy distribution with the probability density function

$f(x;0,1)={\frac {1}{\pi (1+x^{2})}}.\!$

Cumulative distribution function

The cumulative distribution function is:

$F(x;x_{0},\gamma )={\frac {1}{\pi }}\arctan \left({\frac {x-x_{0}}{\gamma }}\right)+{\frac {1}{2}}$

and the inverse cumulative distribution function of the Cauchy distribution is

$F^{-1}(p;x_{0},\gamma )=x_{0}+\gamma \,\tan \left[\pi \,\left(p-{\tfrac {1}{2}}\right)\right].$

Properties

The Cauchy distribution is an example of a distribution which has no mean, variance or higher moments defined. Its mode and median are well defined and are both equal to x₀.

When U and V are two independent normally distributed random variables with expected value 0 and variance 1, then the ratio U/V has the standard Cauchy distribution.

If X₁, …, X_n are independent and identically distributed random variables, each with a standard Cauchy distribution, then the sample mean (X₁ + … + X_n)/n has the same standard Cauchy distribution (the sample median, which is not affected by extreme values, can be used as a measure of central tendency). To see that this is true, compute the characteristic function of the sample mean:

$\phi _{\overline {X}}(t)=\mathrm {E} \left(e^{i\,{\overline {X}}\,t}\right)\,\!$

where ${\overline {X}}$ is the sample mean. This example serves to show that the hypothesis of finite variance in the central limit theorem cannot be dropped. It is also an example of a more generalized version of the central limit theorem that is characteristic of all Lévy skew alpha-stable distributions, of which the Cauchy distribution is a special case.

The Cauchy distribution is an infinitely divisible probability distribution. It is also a strictly stable distribution.

The standard Cauchy distribution coincides with the Student's t-distribution with one degree of freedom.

The location-scale family to which the Cauchy distribution belongs is closed under linear fractional transformations with real coefficients. In this connection, see also McCullagh's parametrization of the Cauchy distributions.

Characteristic function

Let X denote a Cauchy distributed random variable. The characteristic function of the Cauchy distribution is given by

$\phi _{x}(t;x_{0},\gamma )=\mathrm {E} (e^{i\,X\,t})=\exp(i\,x_{0}\,t-\gamma \,|t|).\!$

Why the mean of the Cauchy distribution is undefined

As the Cauchy distribution arises when dividing two independent normal random variables X and Y (taken to have means $\mu _{x}$ and $\mu _{y}$, respectively), one would intuitively assume the mean or expected value to be $\mu _{x}/\mu _{y}$. However, for a standard Cauchy distribution this would result in $0/0$, requiring a more rigorous mathematical treatment:

If a probability distribution has a density function f(x) then the mean is

$\int _{-\infty }^{\infty }xf(x)\,dx.\qquad \qquad (1)\!$

The question is now whether this is the same thing as

$\int _{0}^{\infty }xf(x)\,dx-\int _{-\infty }^{0}|{x}|f(x)\,dx.\qquad \qquad (2)\!$

If at most one of the two terms in (2) is infinite, then (1) is the same as (2). But in the case of the Cauchy distribution, both the positive and negative terms of (2) are infinite. This means (2) is undefined. Moreover, if (1) is construed as a Lebesgue integral, then (1) is also undefined, since (1) is then defined simply as the difference (2) between positive and negative parts.

However, if (1) is construed as an improper integral rather than a Lebesgue integral, then (2) is undefined, and (1) is not necessarily well-defined. We may take (1) to mean

$\lim _{a\to \infty }\int _{-a}^{a}xf(x)\,dx,\!$

and this is its Cauchy principal value, which is zero, but we could also take (1) to mean, for example,

$\lim _{a\to \infty }\int _{-2a}^{a}xf(x)\,dx,\!$

which is not zero, as can be seen easily by computing the integral.

Various results in probability theory about expected values, such as the strong law of large numbers, will not work in such cases.

Why the second moment of the Cauchy distribution is infinite

Without a defined mean, it is impossible to consider the variance or standard deviation of a standard Cauchy distribution. But the second moment about zero can be considered. It turns out to be infinite:

$\mathrm {E} (X^{2})\propto \int _{-\infty }^{\infty }{x^{2} \over 1+x^{2}}\,dx=\int _{-\infty }^{\infty }dx-\int _{-\infty }^{\infty }{1 \over 1+x^{2}}\,dx=\int _{-\infty }^{\infty }dx-\pi =\infty .\!$

Related distributions

• The ratio of two independent standard normal random variables is a standard Cauchy variable, a Cauchy(0,1). Thus the Cauchy distribution is a ratio distribution.
• The standard Cauchy(0,1) distribution arises as a special case of Student's t distribution with one degree of freedom.
• Relation to Lévy skew alpha-stable distribution: if $X\sim {\textrm {Levy-S}}\alpha {\textrm {S}}(1,0,\gamma ,\mu )$ then $X\sim {\textrm {Cauchy}}(\mu ,\gamma ).$

Relativistic Breit-Wigner distribution

In nuclear and particle physics, the energy profile of a resonance is described by the relativistic Breit-Wigner distribution.

See also

• McCullagh's parametrization of the Cauchy distributions
• Lévy flight and Lévy process

External links

• Weisstein, Eric W. "Cauchy Distribution". MathWorld.
• GNU Scientific Library - Reference Manual