Cauchy  (Distribution)

Also known as  "Lorentzian distribution".

Definition of the Cauchy distribution

Let S be an isotropic source emitting particles in the plane, and D a straight line at unit distance from S. Then the impacts of the particles on D are, by definition, distributed as a Cauchy distribution.

In other words, if q is uniformly distributed between -p/2  and + p/2 , then tan(q) is Cauchy distributed by definition.

(See also interactive animation below).

We'll show that the probability density function of the Cauchy distribution is :

Alternate definitions of the Cauchy distribution

Two other classical random variables have Cauchy distributions:

    * The T variable with 1 degree of freedom (see "The t distribution").

    * The ratio of two independent standard normal variables (see animation below and Tutorial).

Cauchy has no moments

The Cauchy distribution is symmetric and bell shaped, just like the normal distribution (see animation below). Therefore, it looks rather innocuous. It is not.

The Cauchy distribution is a classical example of a distribution that has no mean (and consequently no variance or higher order moment) as we show here. As a consequence, the Central Limit Theorem does not apply. In fact, it can be shown that the sample mean always has the same distribution irrespective of the sample size. This distribution is just the original Cauchy distribution (sample size 1).

This pathological behavior is to be contrasted with that of the usual "tame" distributions, like the normal distribution. Think of a scientist who's trying to measure the x position of the source by measuring the positions of a certain number of particle impacts on the screen D. He's considering using the average value of these positions as an estimate of the position of the source. The foregoing result tells us that measuring a single impact or one million impacts will make no difference as to uncertainty relative to the x position of the source.

Translated Cauchy distribution

The probability density function of the translated Cauchy distribution is :

where a is a location parameter.

We describe here the strange behavior of the Best Critical Region of the test pitting :

    * The null hypothesis H₀ : a = 0

    * Against the alternative hypothesis H₁ : a = a₁.

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The following interactive animation illustrates the Cauchy distribution. In particular, it demonstrates the most striking feature of this distribution, namely that the sample mean always has the same distribution, irrespective of the sample size.

Upper frame

    1) The upper frame shows a source S, particle trajectories, and impacts on D (the sample). The emission angles from S are measured with respect to the vertical and are uniformly distributed between - p/2 and + p/2 .

The vertical blue line is the sample mean. It may be that some "impacts" are so far out that they are not within the limits of the frame, and are therefore not shown. Trajectories are then incomplete.

It is even possible that the sample mean be so far out that it is not within the limits of the frame.

You can change the number of "particles" (sample size) with the "Nb Points" spin buttons.

   2) To illustrate the other definition of the Cauchy distribution (ratio of two independent standard variables), click on the "Ratio of Normals" button.

Rather than displaying two normal distributions, we display only one such distribution and draw two independent observations from it (red and blue vertical ticks). The variable that follows a Cauchy distribution is the ratio x₁/x₂  of the x-coordinates of these two observations.

1) What would be the distribution of the inverse ratio x₂ / x₁ ?
2) What would be the distribution of the variable obtained as follows: draw the two observations, then choose randomly between  x₁ / x₂  and
 x₂ / x₁?

Lower frame

The lower frame shows:

    * The standard Cauchy distribution (red),

    * A standard gaussian curve (black). It has no functional role, and is shown just as a reference.

Observe the difference between "Cauchy" and "Gaussian". Both curves look alike (both are symmetrical and bell-shaped), but in fact they are fundamentally different.

    * The Gaussian curve falls off very rapidly as you get away from the central region,

    * Whereas the Cauchy curve has fat, extended "tails". In informal terms, the Cauchy curve falls off just barely fast enough for its integral to be finite (equal to 1). In even more informal terms, the Cauchy curve makes it as a probability distribution, but by as narrow a margin as can be.

Because of this very slow decrease at infinity, the Cauchy distribution has no mean (and therefore no variance or higher order moment). As a consequence, we are now outside the purview of the Central Limit Theorem. We cannot guarantee anymore that the distribution of the sample mean will be normal-like, nor that its variance (assuming it exists) tends toward 0 for large samples. In fact, despite the symmetry of the Cauchy distribution, the sample mean is not a consistent estimator of the distribution median (we refrained from writing "...of the distribution mean", as the distribution has no mean).
 

The main goal of this animation is to illustrate this breakdown of the Central Limit Theorem.

    1) The default option of the animation is "Tangent". Choose "1" for the number of points. Click on "Go" and observe the build-up of the Cauchy distribution.

    2) Click on "Reset", and now choose any number of points. Observe that the red Cauchy curve does not change. Click on "Go". The animation will now build the distribution of the sample mean for samples with the size you chose. You will be rapidly convinced that this distribution does not depend on the sample size. In fact, it is always identical to the original Cauchy distribution.

So the distribution of the sample mean does not become normal for large samples. It has no mean and no variance. Any reasonable measure of "dispersion" of this distribution (e.g. the full width at half maximum) is the same for any sample size.

Some distributions are even more extreme than the Cauchy distribution. For these distributions, the "dispersion" (we cannot speak of variance) of the distribution of the sample mean gets larger as the sample size gets larger.

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The "Ratio of Normals" option has no parameter to be adjusted. Click on "Go", and observe the build up of the ratio of two independent standard normal variables.

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In this Tutorial, we justify three different but equivalent ways to define the Cauchy distribution. Its material comes mainly from the various Tutorials on Transformations of random variables. Most calculations are omitted, as there are given in full detail in these Tutorials.

We also recall that the distribution of the sample mean of the Cauchy distribution is identical to the "mother" distribution.

SOME PROPERTIES OF THE CAUCHY DISTRIBUTION

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Related readings: