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| {{Probability distribution |
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| name =ARGUS|
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| type =density|
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| pdf_image =No image available|
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| cdf_image =No image available|
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| parameters =<math>c > 0</math> cut-off ([[real number|real]])<br />χ > 0 curvature ([[real number|real]])|
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| support =<math>x \in (0, c)\!</math>|
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| pdf =see text|
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| cdf =see text|
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| mean =<math>\mu = c\sqrt{\pi/8}\;\frac{\chi e^{-\frac{\chi^2}{4}} I_1(\tfrac{\chi^2}{4})}{ \Psi(\chi) }</math><br><br> where ''I''<sub>1</sub> is the [[Bessel function|Modified Bessel function]] of the first kind of order 1, and <math>\Psi(x)</math> is given in the text. |
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| median =|
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| mode =<math>\frac{c}{\sqrt2\chi}\sqrt{(\chi^2-2)+\sqrt{\chi^4+4}}</math>|
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| variance =<math>c^2\!\left(1 - \frac{3}{\chi^2} + \frac{\chi\phi(\chi)}{\Psi(\chi)}\right) - \mu^2</math>|
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| skewness =|
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| kurtosis =|
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| entropy =|
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| mgf =|
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| char =|
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| }}
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| {{One source|date=March 2011}}
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| In [[physics]], the '''ARGUS distribution''', named after the [[particle physics]] experiment [[ARGUS (experiment)|ARGUS]],<ref>{{cite doi|10.1016/0370-2693(90)91293-K}} (More formally by the ARGUS Collaboration, H. Albrecht et al.) In this paper, the function has been defined with parameter ''c'' representing the beam energy and parameter ''p'' set to 0.5. The normalization and the parameter χ have been obtained from data.</ref> is the [[probability distribution]] of the reconstructed invariant mass{{Clarify|date=March 2011}} of a decayed particle candidate{{Clarify|date=March 2011}} in continuum background{{Clarify|date=March 2011}}.
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| ==Definition==
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| The [[probability density function]] of the ARGUS distribution is:
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| : <math>
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| f(x; \chi, c ) = \frac{\chi^3}{\sqrt{2\pi}\,\Psi(\chi) }\ \cdot\
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| \frac{x}{c^2} \sqrt{1-\frac{x^2}{c^2}} \
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| \exp\bigg\{ -\frac12 \chi^2\Big(1-\frac{x^2}{c^2}\Big) \bigg\},
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| </math>
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| for {{nowrap|0 ≤ ''x'' < ''c''}}. Here χ, and ''c'' are parameters of the distribution and
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| :<math>\Psi(\chi) = \Phi(\chi)- \chi \phi( \chi ) - \tfrac{1}{2} ,</math> | |
| and Φ(·), ''ϕ''(·) are the [[cumulative distribution function|cumulative distribution]] and [[probability density function]]s of the [[standard normal]] distribution, respectively.
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| == Cumulative distribution function ==
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| The [[cumulative distribution function|cdf]] of the ARGUS distribution is
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| : <math> | |
| F(x) = 1 - \frac{\Psi\Big(\chi\sqrt{1-x^2/c^2}\,\Big)}{\Psi(\chi)}.
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| </math>
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| == Parameter estimation ==
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| Parameter ''c'' is assumed to be known (the [[speed of light]]), whereas ''χ'' can be estimated from the sample ''X''<sub>1</sub>, …, ''X''<sub>''n''</sub> 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
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| : <math>
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| 1 - \frac{3}{\chi^2} + \frac{\chi\phi(\chi)}{\Psi(\chi)} = \frac{1}{n}\sum_{i=1}^n \frac{x_i^2}{c^2}.
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| </math>
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| The solution exists and is unique, provided that the right-hand side is greater than 0.4; the resulting estimator <math style="position:relative;top:-.3em">\scriptstyle\hat\chi</math> is [[consistent estimator|consistent]] and [[asymptotic normality|asymptotically normal]].
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| == Generalized ARGUS distribution ==
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| Sometimes a more general form is used to describe a more peaking-like distribution:
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| : <math>
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| f(x) = \frac{2^{-p}\chi^{2(p+1)}}{\Gamma(p+1)-\Gamma(p+1,\,\tfrac{1}{2}\chi^2)}\ \cdot\
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| \frac{x}{c^2} \bigg( 1 - \frac{x^2}{c^2} \bigg)^p
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| \exp\bigg\{ -\frac12 \chi^2\Big(1-\frac{x^2}{c^2}\Big) \bigg\},
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| \qquad 0 \leq x \leq c,
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| </math>
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| where Γ(·) is the [[gamma function]], and Γ(·,·) is the [[upper incomplete gamma function]].
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| Here parameters ''c'', χ, ''p'' represent the cutoff, curvature, and power respectively.
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| mode = <math>\frac{c}{\sqrt2\chi}\sqrt{(\chi^2-2p-1)+\sqrt{\chi^2(\chi^2-4p+2)+(1+2p)^2}}</math>
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| ''p'' = 0.5 gives a regular ARGUS, listed above.
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| ==References==
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| <references />
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| ==Further reading==
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| *{{cite doi|10.1016/0370-2693(94)01302-0}}
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| *{{cite doi|10.1103/PhysRevLett.107.041803}}
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| *{{cite doi|10.1103/PhysRevLett.104.151802}}
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| {{ProbDistributions|continuous-bounded}}
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| {{Common univariate probability distributions}}
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| [[Category:Continuous distributions]]
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| [[Category:Probability distributions]]
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