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:
:$f(x; \chi, c ) = \frac \cdot \frac \sqrt \exp\bigg\,$
for $0 \leq x < c$. Here $\chi$ and $c$ are parameters of the distribution and

:$\Psi(\chi) = \Phi(\chi)- \chi \phi( \chi ) - \tfrac ,$
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
:$F(x) = 1 - \frac$.

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
:$1 - \frac + \frac = \frac\sum_^n \frac$.

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:
:$f(x) = \frac \cdot \frac \left( 1 - \frac \right)^p \exp\left\, \qquad 0 \leq x \leq c, \qquad c>0,\,\chi>0,\,p>-1$
:$F(x) = \frac, \qquad 0 \leq x \leq c, \qquad c>0,\,\chi>0,\,p>-1$

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:
:$\frac\sqrt$

The mean is:
:$\mu=c \,p \, \sqrt\frac\frac\frac$

where M(·,·,·) is the Kummer's confluent hypergeometric function.Confluent hypergeometric function

The variance is:
:$\sigma^2=c^2 \frac -\mu^2$

''p'' = 0.5 gives a regular ARGUS, listed above.

References

Further reading

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{{ProbDistributions, continuous-bounded

Experimental particle physics
Continuous distributions