Cauchy-Lorentz
Probability density function
The green line is the standard Cauchy distribution |
Cumulative distribution function
Colors match the pdf above |
| Parameters |
Failed to parse (Missing texvc executable; please see math/README to configure.): x_0\!
location (real)
Failed to parse (Missing texvc executable; please see math/README to configure.): \gamma > 0\!
scale (real)
|
| Support |
Failed to parse (Missing texvc executable; please see math/README to configure.): x \in (-\infty; +\infty)\! |
| Probability density function (pdf) |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{\pi\gamma\,\left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]} \! |
| Cumulative distribution function (cdf) |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{\pi} \arctan\left(\frac{x-x_0}{\gamma}\right)+\frac{1}{2} |
| Mean |
not defined |
| Median |
Failed to parse (Missing texvc executable; please see math/README to configure.): x_0 |
| Mode |
Failed to parse (Missing texvc executable; please see math/README to configure.): x_0 |
| Variance |
not defined |
| Skewness |
not defined |
| Excess kurtosis |
not defined |
| Entropy |
Failed to parse (Missing texvc executable; please see math/README to configure.): \ln(4\,\pi\,\gamma)\! |
| Moment-generating function (mgf) |
not defined |
| Characteristic function |
Failed to parse (Missing texvc executable; please see math/README to configure.): \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 broadened by many mechanisms, in particular, collision broadening.
Characterization
Probability density function
The Cauchy distribution has the probability density function
- Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} 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{align}
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 special case when x0 = 0 and γ = 1 is called the standard Cauchy distribution with the probability density function
- Failed to parse (Missing texvc executable; please see math/README to configure.): f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!
Cumulative distribution function
The cumulative distribution function is:
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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, …, Xn are independent and identically distributed random variables, each with a standard Cauchy distribution, then the sample mean (X1 + … + Xn)/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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \phi_{\overline{X}}(t) = \mathrm{E}\left(e^{i\,\overline{X}\,t}\right) \,\!
where Failed to parse (Missing texvc executable; please see math/README to configure.): \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 well defined:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
If a probability distribution has a density function f(x) then the mean or expected value is
- Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\infty}^\infty x f(x)\,dx. \qquad\qquad (1)\!
The question is now whether this is the same thing as
- Failed to parse (Missing texvc executable; please see math/README to configure.): \int_0^\infty x f(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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{a\to\infty}\int_{-a}^a x f(x)\,dx, \!
and this is its Cauchy principal value, which is zero, but we could also take (1) to mean, for example,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{a\to\infty}\int_{-2a}^a x f(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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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 = \infty -\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 Failed to parse (Missing texvc executable; please see math/README to configure.): X\sim \textrm{Levy-S}\alpha\textrm{S}(1,0,\gamma,\mu)
then Failed to parse (Missing texvc executable; please see math/README to configure.): 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
External links
ca:Distribució de Cauchy
cs:Cauchyho rozdělení de:Cauchy-Verteilung es:Distribución de Cauchy fa:توزیع کوشی fr:Loi de Cauchy it:Variabile casuale di Cauchy hu:Cauchy-eloszlás nl:Cauchy-verdeling ja:コーシー分布 pl:Rozkład Cauchy'ego pt:Distribuição de Cauchy ro:Legea de distribuţie Cauchy ru:Распределение Коши su:Sebaran Cauchy
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