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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.
 
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 18:09, 30 June 2013

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

R=X2+Y2.

In terms of the random vector (X,Y), the distribution has the probability density function

f(x,y;α,β)=(β1)(πα2)1[1+(x2+y2α2)]β,

where α and β are seeing dependent parameters. In this form, the distribution is a reparameterisation of a bivariate Student distribution with zero correlation.

In terms of the random variable R, the distribution has density

f(r;α,β)=2rβ1α2[1+(r2α2)]β.

References

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