The Moffat distribution, named after the

physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate caus ...

Anthony Moffat, is a

continuous probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

based upon the

Lorentzian distribution The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...

. Its particular importance in

astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the h ...

is due to its ability to accurately reconstruct point spread functions, whose wings cannot be accurately portrayed by either a

Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...

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=\sqrt.

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

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...

(pdf) :

f(x,y; \alpha,\beta)=\frac\left +\left(\frac\right)\right, \,

where

\alpha

and

\beta

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; \alpha,\beta) = 2 \frac \left +\frac\right . \,

Relation to other distributions

Pearson distribution The Pearson distribution is a family of continuous probability distribution, continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostat ...

Student's t-distribution In probability and statistics, Student's ''t''-distribution (or simply the ''t''-distribution) is any member of a family of continuous probability distributions that arise when estimating the mean of a normally distributed population in sit ...

for

\beta = \frac

Normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...

for

\beta = \frac \rightarrow \infty

, since for the

exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...

\exp x = \lim_ \left(1 + \frac\right)^n.

References

A Theoretical Investigation of Focal Stellar Images in the Photographic Emulsion (1969) – A. F. J. Moffat
{{ProbDistributions, multivariate Continuous distributions Equations of astronomy