Cauchy distribution

FIGURE I Reconciliation results for variable x i with Cauchy distribution and 10% of outliers (, measured value O, reconciled value). 

FIGURE 13 Cauchy distribution case (from Chen et al., 1997). 

The Cauchy distribution, also bell-shaped, is defined over ] — oo, + oo[ and reads... 

If two Cauchy distributions having half-widths Axx and Ax2 are convolved, the result is a similar distribution that has a half-width of Axx + Ax2. If n Cauchy distributions having half-width Axc are convolved, the result is a Cauchy distribution of half-width n Axc. 

One might well ask what the result would be if the Cauchy distribution were convolved with the Gaussian distribution. The result may be obtained from Eqs. (20) and (24) ... 

When simple electrical RC filters are treated, the truncated exponential e, x,H(x) is indispensable. Its transform is given by (2n) 1/2(1 — jco)/( 1 + co2). If the truncated exponential is reflected about the origin, eliminating H(x) and leaving e x, the imaginary part of the transform disappears. We obtain the transform (2/7c)1/2/(l + co2). This is the resonance contour, Cauchy distribution, or Lorentzian shape encountered previously in Section III.B. 

Not all probability distributions have a finite variance a counterexample is the Lorentz or Cauchy distribution... 

The essential difference between the two transition probability densities lies in the fact that for the gaussian distribution pw r, ) the different moments E[Xm], m = 1, 2,. . . , n, exist, while for the Cauchy distribution pc(j, x) they do not exist. The Levy distributions characterized by p(t, k) = exp -a k qT) with 0< <2U 127 128 play a prominent role in the theory of relaxation processes.129 133... 

The Lorentzian peakshape corresponds to a statistical function called the Cauchy distribution. It is less common but often arises in certain types of spectroscopy such as NMR. A simplified equation for a Lorentzian is... 

This condition means Anderson noise of large intensity, and, as we have seen, W is a weak perturbation. Note that on the extreme left and extreme right of the second term of Eq. (41) we have IIL... = — iII[W,...] and (1 — II)L... = — /(I — II) [W,...]. This means that the second term of Eq. (39) is of second order. We aim at illustrating the consequence of making a second-order approximation. To keep our treatment at the second perturbation order, we neglect the perturbation appearing in the exponential of Eq. (41). This makes the calculation of the memory kernel very easy. Using the Cauchy distribution of Eq. (33), we obtain... 

Generally, it is complicated to obtain the distribution F(t) in explicit form, since already Gk consists of convolutions, which can be calculated exphcitly only for the gamma, normal and Cauchy distribution families. Therefore, we will consider a special case to derive some results. 

Computer manufacturer XYZ in the US wants to evaluate the risks from its supplier ABC in Southern China. Inaccurate delivery time is one of the major problems affecting XYZ s supply chain. This represents an N-type risk for XYZ. After analyzing the historical data, XYZ found that ABC s delivery time fits a generalized hyperbolic distribution with parameters X = -0.5, a = 3 = 0, 8 = 1, and p = 0.1. This actually is a special case of generalized hyperbolic distribution called the Cauchy distribution. The PDF and CDF functions of a Cauchy distribution are as follows ... 

In Example 7.2, the performance measure was delivery time which followed a Cauchy Distribution, given by Figure 7.8. The impact function was an N-type Taguchi loss function, representing the cost due to deviation from the target window for delivery time, given by Equation 713. The MtT-N type risk function was given by Equation 714. 

All of the distributions mentioned so far lie within the range of attraction of normal distribution, i.e., the central limit theorem is valid for distributions that are quite dissimilar to the normal distribution such as the discrete Bernoulli distribution and the asymmetric exponential distribution. The Cauchy distribution presented below is, on the contrary, very similar in shape to the normal distribution (see O Fig. 9.18), but it has neither an expected value, nor a (finite) variance, and therefore is exempt from the rule. 

Using the above interpretation, it is easy to convert uniformly distributed 17(—Ji/2, tu/2)random numbers to random numbers with Cauchy distribution. Such a simulated sequence (200 data) is shown in O Fig. 9.19, tt ether with as many normally distributed random numbers having the same FWHM. Note that all of the normal random numbers Ke within a few FWHMs from the origin. The Cauchy-type random numbers, on the other hand, behave in a much more disorderly way, i.e., there are quite a number of points that are way out of the same range. 

Comparison of normai random numbers with random numbers having Cauchy distribution. Both distributions have the same haifwidth (FWHM = 2). Note that some of the Cauchy-type random numbers are as iarge as 500 in this particuiar sequence. On the other hand, aii of the normai random numbers iie in the narrow range of 6er = 5 determined by the 0 3er iimits... 

In Mhssbauer spectroscopy (see Chap. 25 in Vol. 3), the density function of Cauchy distribution is called a Lorentzian curve. In nuclear physics, the same function is also called the Breit-Wigner curve (Lyons 1986). (See also Chap. 2 in this Volume.) This curve is characteristic of the energy uncertainty of excited (nuclear) states, which follows from the fact that excited states have exponential distribution with a finite mean life t. The natural linewidth T, i.e., the FWHM of the Lorentzian energy density, is twice of the parameter y(F = 2y). 

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