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The '''Moffat distribution''', named after the [[physicist]] [[Anthony Moffat]], is a [[continuous probability distribution]] based upon the [[Cauchy distribution|Lorentzian distribution]].  Its particular importance in [[astrophysics]] is due to its ability to accurately reconstruct [[point spread function]]s, whose wings cannot be accurately portrayed by either a [[Gaussian function|Gaussian]] or [[Cauchy distribution|Lorentzian]] function.


==Characterisation==
===Probability density function===
The Moffat distribution can be described in two ways. Firstly as the distribution of a bivariate random variable (''X'',''Y'') centred at zero, and secondly as the distribution of the corresponding radii
:<math>R=\sqrt{X^2+Y^2}.</math>
In terms of the random vector (''X'',''Y''), the distribution has the [[probability density function]]
: <math>
f(x,y; \alpha,\beta)=\left(\beta-1\right)\left(\pi\alpha^2\right)^{-1}\left[1+\left(\frac{x^2+y^2}{\alpha^2}\right)\right]^{-\beta} , \,
</math>


where <math>\alpha</math> and <math>\beta</math> are [[astronomical seeing|seeing]] dependent parameters. In this form, the distribution is a reparameterisation of a [[bivariate Student distribution]] with zero correlation.
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In terms of the random variable ''R'', the distribution has density
: <math>
f(r; \alpha,\beta)=2r \frac{\beta-1}{\alpha^2} \left[1+\left(\frac{r^2}{\alpha^2}\right)\right]^{-\beta} . \,
</math>
 
==References==
 
* [http://adsabs.harvard.edu/abs/1969A%26A.....3..455M A Theoretical Investigation of Focal Stellar Images in the Photographic Emulsion (1969) &ndash; A. F. J. Moffat]
{{ProbDistributions|multivariate}}
[[Category:Continuous distributions]]
[[Category:Probability distributions]]

Revision as of 17:10, 11 February 2014


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