Cauchy Probability Distribution Family

A Cauchy Probability Distribution Family is a continuous probability distribution family that ...

• AKA: Lorentz Distribution.
• See: Scale Parameter, Lévy Distribution, Location Parameter, Real Number, Indeterminate Form, Augustin Cauchy, Continuous Probability Distribution, Hendrik Lorentz, Pathological (Mathematics), Mean, Variance, #Explanation of Undefined Moments.

---

---

References

2016

• (Wikipedia, 2016) ⇒ http://wikipedia.org/wiki/Cauchy_distribution Retrieved:2016-1-28.
  • The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution.

    The Cauchy distribution is often used in statistics as the canonical example of a “pathological” distribution since both its mean and its variance are undefined. (But see the section Explanation of undefined moments below.) The Cauchy distribution does not have finite moments of order greater than or equal to one; only fractional absolute moments exist. The Cauchy distribution has no moment generating function.

    The Cauchy distribution [math]\displaystyle{ f(x; x_0,\gamma) }[/math] is the distribution of the X-intercept of a ray issuing from [math]\displaystyle{ (x_0,\gamma) }[/math] with a uniformly distributed angle. Its importance in physics is the result of it being the solution to the differential equation describing forced resonance. In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane. In spectroscopy, it is the description of the 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 line shape. Many mechanisms cause homogeneous broadening, most notably collision broadening.

    It is one of the few distributions that is stable and has a probability density function that can be expressed analytically, the others being the normal distribution and the Lévy distribution.

2008

• (Upton & Cook, 2008) ⇒ Graham Upton, and Ian Cook. (2008). “A Dictionary of Statistics, 2nd edition revised." Oxford University Press. ISBN:0199541450
  • QUOTE: Cauchy Distribution: A continuous random variable with probability density function f given by [math]\displaystyle{ f(x) = {k \over \pi\{k^2 + (x - m)^2 \}' } - \infty \lt x \lt \infty }[/math] where [math]\displaystyle{ k \gt 0 }[/math] and [math]\displaystyle{ m }[/math] are *parameters is said to have a Cauchy distribution. The graph of f is a bell-curve centred on [math]\displaystyle{ m }[/math]. The mode and the median are both equal to [math]\displaystyle{ m \pm k }[/math], and the *quartiles are m :t ic. A Cauchy distribution has no mean or variance, since, for example, Standard normal distribution l Cauchy distribution. … The Cauchy distribution illustrated has [math]\displaystyle{ m m = 0 }[/math] and [math]\displaystyle{ k = 0.674 }[/math]. Also illustrated is the standard normal distribution. Both distributions have 25% of their area above 0.674 and 25% below - 0.674. The fatter tails of the Cauchy distribution are apparent. [math]\displaystyle{ \int_{-\infty}^{\infty}{kx \over\pi\{k^2 + (x - m)^2\} }dx }[/math] does not exist. The standard Cauchy distribution is given by [math]\displaystyle{ k = 1, m = 0 }[/math] and in this case the distribution is a *[math]\displaystyle{ t }[/math]-distribution, with one *degree of freedom. Since the Cauchy distribution has neither a mean not a variance, the *central limit theorem does not apply. Instead, any linear combination of Cauchy variables has a Cauchy distribution (so that the mean of a random sample of observations from a Cauchy distribution has a Cauchy distribution). If [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] have *independent standard normal distributions then [math]\displaystyle{ Y/X }[/math] has a standard Cauchy distribution. Equivalently, if [math]\displaystyle{ U }[/math] has a *uniform continuous distribution on [math]\displaystyle{ -\frac{1}{2}\pi \lt u \lt \frac{1}{2}\pi }[/math] then tan [math]\displaystyle{ U }[/math] has a standard Cauchy distribution. A geometrical representation of this is as follows. Let [math]\displaystyle{ O }[/math] be the origin of *Cartesian coordinates, and let [math]\displaystyle{ A }[/math] be the point [math]\displaystyle{ (0, 1) }[/math]. If the random point [math]\displaystyle{ P }[/math], with coordinates [math]\displaystyle{ X, 0 }[/math], is such that the angle [math]\displaystyle{ OAP (= u }[/math] say) has a uniform continuous distribution on [math]\displaystyle{ -\frac{1}{2}\pi \lt u \lt \frac{1}{2}\pi }[/math], then [math]\displaystyle{ X }[/math] has a standard Cauchy distribution.

---