Talk:Cauchy distribution

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An application: In a lot of micro-manipulation techniques (Atomic force microscopes, magnetic tweezers, optical tweezers) a harmonic force holds a probe (AFM tip, magnetic/dielectric microsphere) and the probe is subject to thermal noise (white noise). This results in a Lorentzian power-spectrum for the probe motion. Lorentzian fits to measured power spectra are widely used when calibrating these instruments.

—Preceding unsigned comment added by 82.181.221.219 (talk) 18:51, 24 May 2008 (UTC)

Just wondering if someone more knowledgeable might be willing to make this page a little more user-friendly for the interested/educated non-mathematician/non-physicist. Perhaps add some more information on how the Cauchy-Lorenz distribution is used in probability and physics, its applications and implications?

I agree. Start the article with "The Cauchy-Lorenz distribution is used to find solutions to problems involving forced resonance etc. It is defined as <physics equations here>, and has the following properties: <existing article>. Example problem worked through. History (when conceived, first used, etc.)".--203.214.140.61 13:30, 23 July 2006 (UTC)
Or even mention the Cauchy-Lorentz distribution?  ;-P —DIV (128.250.204.118 06:47, 9 July 2007 (UTC))

The nice thing about this distribution is that you can make Cauchy random variables using RND (uniform random variable on [0,1] or the RND and similar functions available in so many environments) and the common tangent function (sine over cosine)

Can someone more knowledgeable check the inverse Cauchy cdf distribution?

When I inverse the CDF I get something different.

Indeed, the inverse CDF has an error, and should be instead x=x0+sigma*tan(pi*(F-1/2)). Anyone mind correcting this?

Done. Thank you for your suggestion! When you feel an article needs changing, please feel free to make whatever changes you feel are needed. Wikipedia is a wiki, so anyone can edit any article by simply following the Edit this page link. You don't even need to log in! (Although there are some reasons why you might like to...) The Wikipedia community encourages you to be bold. Don't worry too much about making honest mistakes—they're likely to be found and corrected quickly. If you're not sure how editing works, check out how to edit a page, or use out the sandbox to try out your editing skills. New contributors are always welcome. --MarkSweep✍ 00:28, 10 September 2005 (UTC)

Contents

[edit] Sample median

The article states "However, the sample median, which is not affected by extreme values, can be used in its place." I think this statement is ambiguous. Does it imply that the sample median can be used as the mean, or that the sample median can be used as the median?. -- Jeff3000 16:32, 29 March 2006 (UTC)

In the current version, this is much clearer: use the sample median instead of the sample mean to report the central tendency of a sample from a Cauchy population (and to estimate the location parameter of the population). --MarkSweep (call me collect) 14:32, 24 July 2006 (UTC)

[edit] Cauchy Equivalent to Student's t-distribution?

I read both here and in print sources (for instance Dagpunar 1988) that a standard Cauchy distribution is a special case of Student's t distribution with one degree of freedom.

How can that be if the Cauchy distribution does not have a defined mean, but S(1) has a mean of zero? I have generated plots and calculated stats for both distributions, and the numbers don't match, either.

There are particularly fast algorithms for generating Cauchy variates, and I would have liked to use that approach for generating S(1) variates, but if the two distributions are really different (as it seems), then I can't do that.

The t distribution has the following pdf:
f(x) = \frac{\Gamma((\nu+1)/2)} {\sqrt{\nu\pi}\,\Gamma(\nu/2)\,(1+x^2/\nu)^{(\nu+1)/2}} \!
Now set ν = 1 and simplify:
f(x) = \frac{\Gamma(1)}{\sqrt{\pi}\, \Gamma(1/2)\, (1+x^2)} \!
Look up the values of the Gamma function. It turns out that Γ(1) = 1 and \Gamma(1/2) = \sqrt{\pi}. Therefore:
f(x) = \frac{1}{\pi\, (1+x^2)} \!
Which is precisely the pdf of the standard Cauchy distribution. --MarkSweep (call me collect) 17:59, 23 July 2006 (UTC)

[edit] HWHM versus FWHM

Are you sure, that γ is the half width at half maximum (HWHM) and not full width at half maximum (FWHM)? In articles in other languages I found that it is FWHM. ...? 149.220.35.153 09:11, 26 January 2007 (UTC)Len

Those other articles are wrong then. Forget about the offset x0. The Cauchy distribution is
f=\frac{1}{\pi}\left(\frac{\gamma}{x^2+\gamma^2}\right)
Its pretty easy to see that when x=0, f is at its maximum value of 1/πγ and when x=γ f is half of that. So that means the half width at half maximum is γ and the full width at half maximum will be twice that or 2γ. PAR 15:06, 26 January 2007 (UTC)

Thank you very much. Now I could find where my mistake was. In all the other articles was written

f(x) = \frac{1}{2\pi}\frac{\Gamma}{x^2+\Gamma^2/4}

and with that, Gamma is FWHM, of course. 149.220.35.153 13:26, 29 January 2007 (UTC)LEN

In the reference to central limit theorem, it is mentioned that the sample mean (X1+...+Xn)/n has the same distribution as Xi, so central limit thm doesn't apply. But doesn't central limit theorem deal with (X1+..+Xn)/sqrt(n)?

65.219.4.5 01:38, 10 March 2007 (UTC)corecirculator

[edit] Is it correct to redirect Lorentzian function to this side???

I was looking for the definition of the Lorentzian function and got a little confused when I was redirected here. OK the probability function of the Cauchy distribution follows a Lorentzian function (I know that now) but if you don't know that from the begining you don't know what to look for. For example I was reading a text which said that a focused telescope will receive backscatter as a Lorentzian function of the distance centered around the focus distance. Turning up at a distribution page wasn't really what I expected ...

A "Lorentzian function" google search gives 70 000 hits and Gaussian function has a (not so good?) wikipedia page so I think the Lorentzian can deserves a page of its own. Otherwise I would like to say that the people behind the distribution pages on wikipedia has and are doing a great job!!!!

Petter

Hi - "Lorentzian function" is just another name for "Cauchy distribution", so it definitely should not have a separate page. I have put in that it is sometimes known as the Lorentzian function. PAR 15:51, 31 May 2007 (UTC)