Cauchy Probability Distribution Family

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



References

2016

  1. http://webphysics.davidson.edu/Projects/AnAntonelli/node5.html Note that the intensity, which follows the Cauchy distribution, is the square of the amplitude.

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.