# ARGUS distribution

In physics, the ARGUS distribution, named after the particle physics experiment ARGUS, is the probability distribution of the reconstructed invariant mass of a decayed particle candidate in continuum background.

## Definition

The probability density function (pdf) of the ARGUS distribution is:
    $f(x;\chi ,c)={\frac {\chi ^{3}}{{\sqrt {2\pi }}\,\Psi (\chi )}}\cdot {\frac {x}{c^{2}}}{\sqrt {1-{\frac {x^{2}}{c^{2}}}}}\exp {\bigg \-{\frac {1}{2}}\chi ^{2}{\Big1-{\frac {x^{2}}{c^{2}}}{\Big )}{\bigg \}},$
for $0\leq x<c$. Here $\chi$ and $c$ are parameters of the distribution and
    $\Psi (\chi )=\Phi (\chi )-\chi \phi (\chi )-{\tfrac {1}{2}},$
where $\Phi (x)$ and $\phi (x)$ are the cumulative distribution and probability density functions of the standard normal distribution, respectively.

## Cumulative distribution function

The cumulative distribution function (cdf) of the ARGUS distribution is
    $F(x)=1-{\frac {\Psi \left(\chi {\sqrt {1-x^{2}/c^{2}}}\right)}{\Psi (\chi )}}$.

## Parameter estimation

Parameter c is assumed to be known (the kinematic limit of the invariant mass distribution), whereas χ can be estimated from the sample X$_1$, …, X$_n$ using the maximum likelihood approach. The estimator is a function of sample second moment, and is given as a solution to the non-linear equation
    $1-{\frac {3}{\chi ^{2}}}+{\frac {\chi \phi (\chi )}{\Psi (\chi )}}={\frac {1}{n}}\sum _{i=1}^{n}{\frac {x_{i}^{2}}{c^{2}}}$.
The solution exists and is unique, provided that the right-hand side is greater than 0.4; the resulting estimator $\scriptstyle {\hat {\chi }}$ is consistent and asymptotically normal.

## Generalized ARGUS distribution

Sometimes a more general form is used to describe a more peaking-like distribution:
    $f(x)={\frac {2^{-p}\chi ^{2(p+1)}}{\Gamma (p+1)-\Gamma (p+1,\,{\tfrac {1}{2}}\chi ^{2})}}\cdot {\frac {x}{c^{2}}}\left(1-{\frac {x^{2}}{c^{2}}}\right)^{p}\exp \left\{-{\frac {1}{2}}\chi ^{2}\left(1-{\frac {x^{2}}{c^{2}}}\right)\right\},\qquad 0\leq x\leq c,\qquad c>0,\,\chi >0,\,p>-1$
    $F(x)={\frac {\Gamma \left(p+1,\,{\tfrac {1}{2}}\chi ^{2}\left(1-{\frac {x^{2}}{c^{2}}}\right)\right)-\Gamma (p+1,\,{\tfrac {1}{2}}\chi ^{2})}{\Gamma (p+1)-\Gamma (p+1,\,{\tfrac {1}{2}}\chi ^{2})}},\qquad 0\leq x\leq c,\qquad c>0,\,\chi >0,\,p>-1$
where Γ(·) is the gamma function, and Γ(·,·) is the upper incomplete gamma function.
Here parameters c, χ, p represent the cutoff, curvature, and power respectively.
The mode is:
    ${\frac {c}{{\sqrt {2}}\chi }}{\sqrt {(\chi ^{2}-2p-1)+{\sqrt {\chi ^{2}(\chi ^{2}-4p+2)+(1+2p)^{2}}}}}$
The mean is:
    $\mu =c\,p\,{\sqrt {\pi }}{\frac {\Gamma (p)}{\Gamma ({\tfrac {5}{2}}+p)}}{\frac {\chi ^{2p+2}}{2^{p+2}}}{\frac {M\left(p+1,{\tfrac {5}{2}}+p,-{\tfrac {\chi ^{2}}{2}}\right)}{\Gamma (p+1)-\Gamma (p+1,\,{\tfrac {1}{2}}\chi ^{2})}}$
where M(·,·,·) is the Kummer's confluent hypergeometric function.
The variance is:
    $\sigma ^{2}=c^{2}{\frac {\left({\frac {\chi }{2}}\right)^{p+1}\chi ^{p+3}e^{-{\tfrac {\chi ^{2}}{2}}}+\left(\chi ^{2}-2(p+1)\right)\left\{\Gamma (p+2)-\Gamma (p+2,\,{\tfrac {1}{2}}\chi ^{2})\right\}}{\chi ^{2}(p+1)\left(\Gamma (p+1)-\Gamma (p+1,\,{\tfrac {1}{2}}\chi ^{2})\right)}}-\mu ^{2}$
p = 0.5 gives a regular ARGUS, listed above.