|Probability density function|
The green line is the standard Cauchy distribution
|Cumulative distribution function|
Cumulative distribution function for the Normal distribution
Colors match the pdf above
|Parameters|| location (real)|
|Probability density function (pdf)|
|Cumulative distribution function (cdf)|
|Excess kurtosis||(not defined)|
|Moment-generating function (mgf)||(not defined)|
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.
- 1 Characterization
- 2 Properties
- 3 Why the mean of the Cauchy distribution is undefined
- 4 Why the second moment of the Cauchy distribution is infinite
- 5 Related distributions
- 6 Relativistic Breit-Wigner distribution
- 7 See also
- 8 External links
Probability density function
The Cauchy distribution has the probability density function
The special case when x0 = 0 and γ = 1 is called the standard Cauchy distribution with the probability density function
Cumulative distribution function
The cumulative distribution function is:
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.
Let X denote a Cauchy distributed random variable. The characteristic function of the Cauchy distribution is well defined:
Why the mean of the Cauchy distribution is undefined
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.
The general form of the Cauchy percent point function is: G(p) = t–(s) cot(pi*p)
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:
- 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 then .