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Cauchy distribution

Probability Density The green curve corresponds to the standard Cauchy distribution
distribution function The colors are in accordance with the chart above
Designation	$\mathrm(C)(x_0,\gamma)$
Options	$x_0$ - shift factor $\gamma > 0$ - scale factor
Carrier	$x \in (-\infty; +\infty)$
Probability Density	$\frac(1)(\pi\gamma\,\left)$
distribution function	$\frac(1)(\pi) \mathrm(arctg)\left(\frac(x-x_0)(\gamma)\right)+\frac(1)(2)$
Expected value	does not exist
Median	$x_0$
Fashion	$x_0$
Dispersion	$+\infty$
Asymmetry coefficient	does not exist
Kurtosis coefficient	does not exist
Differential entropy	$\ln(4\,\pi\,\gamma)$
Generating function of moments	not determined
characteristic function	$\exp(x_0\,i\,t-\gamma\,$

Definition

Let the distribution of a random variable

X

given by density

f_X(x)

, having the form:

$f_X(x) = \frac(1)(\pi\gamma \left) = ( 1 \over \pi ) \left[ ( \gamma \over (x - x_0)^2 + \gamma^2 ) \right]$ ,

$x_0 \in \mathbb(R)$ - shift parameter;
$\gamma > 0$ - scale parameter.

Then they say that $X$ has a Cauchy distribution and write $X \sim \mathrm(C)(x_0,\gamma)$ . If $x_0 = 0$ And $\gamma = 1$ , then this distribution is called standard Cauchy distribution.

distribution function

F^(-1)_X(x) = x_0 + \gamma\,\mathrm(tg)\,\left[\pi\,\left(x-(1 \over 2)\right)\right].

This allows a sample to be generated from the Cauchy distribution using the inverse transform method.

Moments

\int\limits_(-\infty)^(\infty)\!x^(\alpha)f_X(x)\, dx

not defined for $\alpha \geqslant 1$ , nor the mathematical expectation (although the integral of the 1st moment in the sense of the main value is equal to: $\lim\limits_(c \rightarrow \infty) \int\limits_(-c)^(c) x \cdot ( 1 \over \pi ) \left[ ( \gamma \over (x - x_0)^2 + \ gamma^2 ) \right]\, dx = x_0$ ), neither the variance nor the higher-order moments of this distribution have been determined. It is sometimes said that the mathematical expectation is not defined and the variance is infinite.

Other properties

The Cauchy distribution is infinitely divisible.
The Cauchy distribution is stable. In particular, the sample mean of a sample from a standard Cauchy distribution itself has a standard Cauchy distribution: if $X_1,\ldots, X_n \sim \mathrm(C)(0,1)$ , That

\overline(X) = \frac(1)(n) \sum\limits_(i=1)^n X_i \sim \mathrm(C)(0,1)

Relationship with other distributions

If $U \sim U$ , That

x_0 + \gamma\,\mathrm(tg)\,\left[\pi\left(U-(1 \over 2)\right)\right] \sim \mathrm(C)(x_0,\gamma)

If $X_1, X_2$ are independent normal random variables such that $X_i \sim \mathrm(N)(0,1),\; i=1,2$ , That

\frac(X_1)(X_2) \sim \mathrm(C)(0,1)

The standard Cauchy distribution is a special case of Student's distribution:

\mathrm(C)(0,1) \equiv \mathrm(t)(1)

Appearance in practical problems

The Cauchy distribution characterizes the length of the segment cut off on the abscissa by a straight line fixed at a point on the y-axis if the angle between the line and the y-axis has a uniform distribution on the interval (−π; π) (i.e. the direction of the line is isotropic on the plane).
In physics, the Cauchy distribution (also called the Lorentz form) describes the profiles of uniformly broadened spectral lines.
The Cauchy distribution describes the amplitude-frequency characteristics of linear oscillatory systems in the vicinity of resonant frequencies.

	P Probability distributions
	One-dimensional	Multidimensional
Discrete:	Bernoulli \| Binomial \| Geometric \| Hypergeometric \| Logarithmic \| Negative binomial \| Poisson \| Discrete Uniform	Multinomial
Absolutely continuous:	Beta \| Weibulla \| Gamma \| Hyperexponential \| Gompertz distribution \| Kolmogorov \| Cauchy\| Laplace \| Lognormal \| Normal (Gaussian) \| Logistics \| Nakagami \| Pareto \| Pearson \| \| Exponential \| Variance-gamma	Multidimensional normal \| copula

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An excerpt characterizing the Cauchy Distribution

Rostov gave the spurs to his horse, called out to non-commissioned officer Fedchenko and two more hussars, ordered them to follow him, and rode at a trot downhill in the direction of the continuing screams. Rostov was both terribly and merry to go alone with three hussars there, to this mysterious and dangerous foggy distance, where no one had been before him. Bagration shouted to him from the mountain so that he would not go further than the stream, but Rostov pretended not to hear his words, and, without stopping, rode on and on, constantly deceived, mistaking bushes for trees and potholes for people and constantly explaining his deceptions. Having trotted downhill, he no longer saw either ours or the enemy's fires, but he heard the cries of the French louder and clearer. In the hollow he saw something like a river in front of him, but when he reached it, he recognized the road he had traveled. Riding out onto the road, he held his horse back, undecided whether to ride on it or cross it and ride uphill across the black field. It was safer to drive along the road brightened in the fog, because people could be seen more quickly. “Follow me,” he said, crossed the road and began to gallop up the mountain, to the place where the French picket had been standing since evening.
“Your Honor, here it is!” one of the hussars spoke from behind.
And before Rostov had time to make out something suddenly blackened in the fog, a light flashed, a shot clicked, and the bullet, as if complaining about something, buzzed high in the fog and flew out of hearing. The other gun did not fire, but a light flashed on the shelf. Rostov turned his horse and galloped back. Another four shots rang out at different intervals, and bullets sang in different tones somewhere in the fog. Rostov reined in his horse, which had cheered up just as much as he did from the shots, and rode off at a pace. "Well, more, well, more!" a cheerful voice spoke in his soul. But there were no more shots.
Just approaching Bagration, Rostov again put his horse into a gallop and, holding his hand at the visor, rode up to him.
Dolgorukov kept insisting on his opinion that the French had retreated and only in order to deceive us they had put out fires.
– What does this prove? - he said at the time when Rostov drove up to them. “They could have retreated and left the pickets.
- Apparently, not everyone has left yet, prince, - said Bagration. Until tomorrow morning, we'll find out tomorrow.
“There is a picket on the mountain, Your Excellency, everything is the same as it was in the evening,” Rostov reported, leaning forward, holding his hand at the visor and unable to restrain the smile of fun caused in him by his trip and, most importantly, by the sounds of bullets.
“Good, good,” said Bagration, “thank you, Mr. Officer.
“Your Excellency,” said Rostov, “permit me to ask you.
- What's happened?
- Tomorrow our squadron is assigned to the reserves; let me ask you to attach me to the 1st squadron.
- What's your last name?
- Count Rostov.
- Oh good. Stay with me as an orderly.
- Ilya Andreich's son? Dolgorukov said.
But Rostov did not answer him.
“So I hope, Your Excellency.
- I'll order.
“Tomorrow, very possibly, they will send some kind of order to the sovereign,” he thought. - God bless".

The cries and fires in the enemy army came from the fact that while the order of Napoleon was being read to the troops, the emperor himself was riding around his bivouacs. The soldiers, seeing the emperor, lit bunches of straw and, shouting: vive l "empereur!, ran after him. Napoleon's order was as follows:
"Soldiers! The Russian army comes out against you to avenge the Austrian, Ulm army. These are the same battalions which you defeated at Gollabrunn and which you have been constantly pursuing to this place ever since. The positions we occupy are powerful, and as long as they go to get around me on the right, they will expose me to the flank! Soldiers! I myself will lead your battalions. I will keep far from the fire if you, with your usual courage, bring disorder and confusion into the ranks of the enemy; but if victory is even for a moment doubtful, you will see your emperor exposed to the first blows of the enemy, because there can be no hesitation in victory, especially on a day when the honor of the French infantry, which is so necessary for the honor of his nation, is at stake.

Physical Encyclopedia

CAUCHI DISTRIBUTION

Probability distribution with density

and distribution function

Shift parameter, >0 - scale parameter. Reviewed in 1853 by O. Cauchy. characteristic function K. r. equal to exp ; moments of order R 1 do not exist, so law of large numbers for K. r. fails [if X 1 ..., X n are independent random variables with the same K. r., then n -1 (X 1 + ... + X n) has the same K. r.]. Family K. r. closed under linear transformations: if the random variable X has a distribution (*), then aX+b also has K. r. with parameters , . K. r.- sustainable distribution with exponent 1, symmetrical about a point x=. K. r. has, for example, the relationship X/Y independent normally distributed random variables with zero means, as well as a function , where the random variable Z evenly distributed over . They also consider multidimensional analogs of K. r.

Lit.: V. Feller, Introduction to probability theory and its applications, trans. from English, vol. 2, M., 1984.

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For the Cauchy distribution, there is not even the first initial moment of the distribution, that is, the mathematical expectation, since the integral defining it diverges. In this case, the distribution has both a median and a mode, which are equal to the parameter a.

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It follows from the above that the approximation by the Cauchy distribution of random processes , which are characterized by a finite mathematical expectation and a finite variance, is invalid.

So, we have obtained a symmetrical distribution depending on three parameters, which can be used to describe samples of random variables, including those with gentle slopes. However, this distribution has disadvantages, which were considered when discussing the Cauchy distribution, namely, the mathematical expectation exists only for a > 1, the variance is finite only for OS > 2, and in general, the final moment of the k-th order distribution exists for a > k .

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