The green line is the standard Cauchy distribution|
Colors match the pdf above|
support =|pdf =| cdf = |
mean =not defined|
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char =}} 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.
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
and the inverse cumulative distribution function of the Cauchy distribution is
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:
where 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 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.
As the Cauchy distribution arises when dividing two independent normal random variables X and Y (taken to have means and , respectively), one would intuitively assume the mean or expected value to be . However, for a standard Cauchy distribution this would result in , requiring a more rigorous mathematical treatment:
The question is now whether this is the same thing as
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.
and this is its Cauchy principal value, which is zero, but we could also take (1) to mean, for example,
which is not zero, as can be seen easily by computing the integral.
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: