Moffat distribution

The Moffat distribution, named after the

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

$${\displaystyle R={\sqrt {X^{2}+Y^{2}}}.}$$

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

$${\displaystyle f(x,y;\alpha ,\beta )={\frac {\beta -1}{\pi \alpha ^{2}}}\left[1+\left({\frac {x^{2}+y^{2}}{\alpha ^{2}}}\right)\right]^{-\beta },}$$

${\displaystyle \alpha }$

${\displaystyle \beta }$

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

$${\displaystyle f(r;\alpha ,\beta )=2{\frac {\beta -1}{\alpha ^{2}}}\left[1+{\frac {r^{2}}{\alpha ^{2}}}\right]^{-\beta }.}$$

Relation to other distributions

• Pearson distribution
• Student's t-distribution for ${\displaystyle \beta ={\frac {\alpha ^{2}+1}{2}}}$
• Normal distribution for ${\displaystyle \beta ={\frac {\alpha ^{2}}{2}}\rightarrow \infty }$, since for the exponential function ${\displaystyle \exp x=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}.}$

References

• A Theoretical Investigation of Focal Stellar Images in the Photographic Emulsion (1969) – A. F. J. Moffat