Cauchy's law of distribution of random variables. Kosha distribution

It would seem that the Cauchy distribution looks very attractive for describing and modeling random variables. However, in reality this is not the case. The properties of the Cauchy distribution are sharply different from the properties of the Gaussian, Laplace and other exponential distributions.

The fact is that the Cauchy distribution is close to extremely flat. Recall that a distribution is said to be extremely flat if, as x -> +oo, its probability density

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

Of course, the dispersion of this distribution (the second central moment) is also equal to infinity. In practice, this means that the estimate of variance for a sample from the Cauchy distribution will increase without limit as the volume of data increases.

From the above it follows that approximation by the Cauchy distribution of random processes, which are characterized by finite mathematical expectation and finite variance, is incorrect.

So, we have obtained a symmetric distribution depending on three parameters, with the help of which we can describe samples of random variables, including those with gentle slopes. However, this distribution has disadvantages that 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 finite moment of the kth order distribution exists for a > k .

Figure 14.1 uses 8,000 samples from the famous Cauchy distribution, which has an infinite mean and variance. The Cauchy distribution is described in more detail below. The series used here was "normalized" by subtracting the mean and dividing by the sample standard deviation. Thus, all units are expressed in standard deviations. For comparison, we use 8,000 Gaussian random variables that have been normalized in a similar way. It's important to understand that the next two steps will always end up with a mean of 0 and a standard deviation of 1 because they were normalized to those values. Convergence means that the time series quickly moves towards a stable value.

These two well-known distributions, the Cauchy distribution and the normal distribution, have many applications. They are also the only two members of the family of stable distributions for which probability density functions can be derived explicitly. In all other fractional cases they must be estimated, usually by numerical means. We will discuss one of these methods in a later section of this chapter.

In Chapter 14, we examined the serial standard deviation and mean of the American stock market and compared it with a time series derived from the Cauchy distribution. We did this to see the effect of infinite variance and mean on the time series. Serial standard deviation is the standard deviation of a time series when we add at a time

Make a first approximation of Z to u(o,F) by taking the weighted average of the F quantiles of the Cauchy and Gaussian distributions.

Table A3.2 converts the results of Table A3.1 into quantiles. To find out which F value explains 99 percent of the observations for a = 1.0, move down the F column to the left to 0.99 and across to u = 31.82. The Cauchy distribution requires observations 31.82 values from the mean to cover 99 percent probability. In contrast, the normal case reaches the 99 percent level at u=3.29. This differs from the standard normal case, which is 2.326 standard deviations rather than 3.29 s.

P(> (nm)1/2Г(n/2) n When n = 1, the corresponding distribution is called the Cauchy distribution.

If a series is stationary in the broad sense, then it is not necessarily strictly stationary. At the same time, a strictly stationary series may not be stationary in the broad sense simply because it may not have a mathematical expectation and/or dispersion. (In relation to the latter, an example would be a random sample from the Cauchy distribution.) In addition, situations are possible when the above three conditions are met, but, for example, E(X) depends on t.

At the same time, in the general case, even if some random variables X, . .., X are mutually independent and have the same distribution, this does not mean that they form a white noise process, because the random variable Xt may simply not have a mathematical expectation and/or variance (we can again point to the Cauchy distribution as an example).

When two or more factors, for example labor and material assets, are involved in the process of production of goods and provision of services, as well as in the subsequent formation of cash receipts, a logical distribution of the latter among factors seems generally impossible. It was assumed that the assets that could be used would be matched with net marginal revenues, but the amount of private marginal revenues may turn out to be higher than the total net revenues from the sale of products and the provision of services.

Such long-tailed distributions, especially in Pareto data, led Levy (1937), a French mathematician, to formulate the generalized density function, of which normal distributions as well as Cauchy distributions were special cases. Levy used a generalized version of the Central Limit Theorem. These distributions correspond to a large class of natural phenomena, but they have not received much attention due to their unusual and seemingly intractable problems. Their unusual properties continue to make them unpopular, but their other properties are so close to our results from capital markets that we must explore them. In addition, stable Lévy distributions have been found to be useful in describing the statistical properties of turbulent flow and l/f noise - and they are also fractal.

Figure 14.2(a) shows the serial standard deviation for those two series. Serial standard deviation, like serial average, is a calculation of the standard deviation as observations are added one at a time. In this case the difference is even more striking. The random ejad quickly converges to a standard deviation of 1. The Cauchy distribution, in contrast, never converges. Instead, it is characterized by several large intermittent jumps and large deviations from the normalized value of 1.

This is the logarithm of the characteristic function for the Cauchy distribution, which is known to have infinite variance and mean. In this case, 8 becomes the median of the distribution, and c becomes the seven-interquartile range.

Holt and Row (1973) found probability density functions for a = 0.25 to 2.00 and P equal to -1.00 to +1.00, both in increments of 0.25. The methodology they used interpolated between known distributions, such as Cauchy and normal distributions, and the integral representation from the work of Zolotarev (1964/1966). Tables prepared for the former

As we discussed in Chapter 14, explicit expressions for stable distributions exist only for special cases of normal and Cauchy distributions. However, Bergstrom (1952) developed a series expansion that Fame and Roll used to approximate densities for many values of alpha. When a > 1.0, they could use Bergstrom's results to derive the next convergent series

Material from Wikipedia - the free encyclopedia

Cauchy distribution

Probability Density The green curve corresponds to the standard Cauchy distribution
Distribution function The colors are according to the chart above
Designation	$\mathrm(C)(x_0,\gamma)$
Options	$x_0$ - shift coefficient $\gamma > 0$ - 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 is written $X \sim \mathrm(C)(x_0,\gamma)$ . If $x_0 = 0$ And $\gamma = 1$ , then such a 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 principal 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 dispersion nor the higher order moments of this distribution are determined. Sometimes they say that the mathematical expectation is undefined, but 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 the Student 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 x-axis of a straight line fixed at a point on the ordinate axis, if the angle between the straight line and the ordinate axis has a uniform distribution on the interval (−π; π) (i.e., the direction of the straight 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 \| Weibull \| Gamma \| Hyperexponential \| Gompertz distribution \| Kolmogorov \| Cauchy\| Laplace \| Lognormal \| Normal (Gaussian) \| Logistics \| Nakagami \| Pareto \| Pearson \| \| Exponential \| Variance-gamma	Multivariate normal \| Copula

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An excerpt characterizing the Cauchy Distribution

Rostov gave spurs to his horse, called out to non-commissioned officer Fedchenka and two more hussars, ordered them to follow him and trotted down the hill towards the continued screams. It was both scary and fun for Rostov to travel alone with three hussars there, into this mysterious and dangerous foggy distance, where no one had been before. Bagration shouted to him from the mountain so that he should not go further than the stream, but Rostov pretended as if he had not heard his words, and, without stopping, rode further and further, constantly being deceived, mistaking bushes for trees and potholes for people and constantly explaining his deceptions. Trotting down the mountain, he no longer saw either ours or the enemy’s fires, but heard the cries of the French louder and more clearly. In the hollow he saw in front of him something like a river, but when he reached it, he recognized the road he had passed. Having ridden out onto the road, he reined in his horse, undecided: to ride along it, or to cross it and ride uphill through a black field. It was safer to drive along the road that became lighter in the fog, because it was easier to see people. “Follow me,” he said, crossed the road and began to gallop up the mountain, to the place where the French picket had been stationed since the evening.
- Your Honor, here he is! - one of the hussars said from behind.
And before Rostov had time to see 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 earshot. The other gun did not fire, but a light flashed on the shelf. Rostov turned his horse and galloped back. Four more shots rang out at different intervals, and bullets sang in different tones somewhere in the fog. Rostov reined in his horse, which was as cheerful as he was from the shots, and rode at a walk. “Well then, well again!” some 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 still insisted on his opinion that the French had retreated and only set up the fires to deceive us.
– What does this prove? - he said as Rostov drove up to them. “They could retreat and leave the pickets.”
“Apparently, not everyone has left yet, prince,” said Bagration. – Until tomorrow morning, tomorrow we’ll find out everything.
“There’s a picket on the mountain, your Excellency, still in the same place where it was in the evening,” Rostov reported, bending forward, holding his hand to the visor and unable to contain the smile of amusement caused in him by his trip and, most importantly, by the sounds of bullets.
“Okay, okay,” said Bagration, “thank you, Mr. Officer.”
“Your Excellency,” said Rostov, “allow me to ask you.”
- What's happened?
“Tomorrow our squadron is assigned to reserves; Let me ask you to second me to the 1st squadron.
- What's your last name?
- Count Rostov.
- Oh good. Remain with me as an orderly.
– Ilya Andreich’s son? - said Dolgorukov.
But Rostov did not answer him.
- So I will hope, Your Excellency.
- I will order.
“Tomorrow, perhaps, they will send some kind of order to the sovereign,” he thought. - God bless".

The screams and fires in the enemy army occurred because while Napoleon's order was being read among the troops, the emperor himself was riding around his bivouacs on horseback. 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 that you defeated at Gollabrunn and which you have since constantly pursued to this place. The positions we occupy are powerful, and while they move to flank me on the right, they will expose my flank! Soldiers! I myself will lead your battalions. I will stay far from the fire if you, with your usual courage, bring disorder and confusion into the enemy’s ranks; but if victory is in doubt for even one minute, you will see your emperor exposed to the first blows of the enemy, because there can be no doubt in victory, especially on a day in which the honor of the French infantry, which is so necessary for the honor of his nation, is at issue.

CAUCHY DISTRIBUTION, probability distribution of a random variable X having density

where - ∞< μ < ∞ и λ>0 - parameters. The Cauchy distribution is unimodal and symmetric relative to the point x = μ, which is the mode and median of this distribution [Figures a and b show graphs of the density p(x; λ, μ) and the corresponding distribution function F (x; λ, μ) for μ =1 ,5 and λ = 1]. The mathematical expectation of the Cauchy distribution does not exist. The characteristic function of the Cauchy distribution is equal to e iμt - λ|t| , - ∞< t < ∞. Произвольное Коши распределение с параметрами μ и λ выражается через стандартное Коши распределение с параметрами 0 и 1 формулой

If independent random variables X 1,...,X n have the same Cauchy distribution, then their arithmetic mean (X 1 + ... + X n)/n for any n = 1,2, ... has same distribution; this fact was established by S. Poisson (1830). The Cauchy distribution is a stable distribution. The ratio X/Y of independent random variables X and Y with a standard normal distribution has a Cauchy distribution with parameters 0 and 1. The distribution of the tangent tan Z of a random variable Z, with a uniform distribution on the interval [-π/2, π/2], also has a Cauchy distribution distribution with parameters 0 and 1. The Cauchy distribution was considered by O. Cauchy (1853).

Physical encyclopedia

CAUCHY 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 don't exist, so law of large numbers for K. r. not executed [if X 1 ..., Xn are independent random variables with the same K. r., then n -1 (X 1 + ... + X n) has the same K. r.]. Family K. b. closed under linear transformations: if the random variable X has distribution (*), then aX+b also has K. r. with parameters , . K.r.- sustainable distribution with exponent 1, symmetrical about the point x=. K.r. has, for example, the relation X/Y independent normally distributed random variables with zero means, as well as the function , where the random variable Z evenly distributed over . Multidimensional analogues of K. r. are also considered.

Lit.: Feller V., Introduction to Probability Theory and Its Applications, trans. from English, vol. 2, M., 1984.

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