Dear all,(adsbygoogle = window.adsbygoogle || []).push({});

I'm not a mathematician so please excuse me for a certain lack of strictness ...

I work on random signals in physics, these signals are most of the time called "noise" for us. For example, we can speak about x(t), a time domain random signal.

Very usually, the statistics for noise is gaussian or poissonian, and so we use to speak of the histogram of this signal, in terms of second order moment or variance : it's practical because this single number permits to compare various noises.

Unfortunately, recently I started to work on signals which histogram is a lorentzian shape. If I understood, this corresponds to the Cauchy distribution. And again, if I understood, the second order momentum is inifinite : when I try to solve the second order moment integral, I obtain the infinite ... not practical. And indeed, a Lorentzian width is usually known thanks to its full with half maximum and not by its variance (before calculating the variance, I thought this choice was arbitrary ;-) ).

Moreover, things can become more complex as some histograms can be voigt-function shaped (= lorentzian-gaussian convolution).

So here is my question : I would like to compare the histograms of all my noises, whatever their statistics, by speaking of their "range" (= variance for a gaussian shape for example). I understand it is not possible to use the second order momentum for that purpose. Is there some other way to compare the range of these statistics ?

Perhaps my question does not makes sense for some fundamental reason, if this is the case I would like to understand why.

Best Regards,

Mike

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# Variance and Cauchy Distribution

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