ARGUS distribution

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ARGUS
Probability density function No image available
Cumulative distribution function No image available
Parameters	$c>0$ cut-off (real) χ > 0 curvature (real)
Support	$x\in (0,c)\!$
pdf	see text
CDF	see text
Mean	$\mu =c{\sqrt {\pi /8}}\;{\frac {\chi e^{{-{\frac {\chi ^{2}}{4}}}}I_{1}({\tfrac {\chi ^{2}}{4}})}{\Psi (\chi )}}$ where I₁ is the Modified Bessel function of the first kind of order 1, and $\Psi (x)$ is given in the text.
Mode	${\frac {c}{{\sqrt 2}\chi }}{\sqrt {(\chi ^{2}-2)+{\sqrt {\chi ^{4}+4}}}}$
Variance	$c^{2}\!\left(1-{\frac {3}{\chi ^{2}}}+{\frac {\chi \phi (\chi )}{\Psi (\chi )}}\right)-\mu ^{2}$

In physics, the ARGUS distribution, named after the particle physics experiment ARGUS,^[1] is the probability distribution of the reconstructed invariant mass of a decayed particle candidate in continuum background.

Definition

The probability density function of the ARGUS distribution is:

$f(x;\chi ,c)={\frac {\chi ^{3}}{{\sqrt {2\pi }}\,\Psi (\chi )}}\ \cdot \ {\frac {x}{c^{2}}}{\sqrt {1-{\frac {x^{2}}{c^{2}}}}}\ \exp {\bigg \{}-{\frac 12}\chi ^{2}{\Big (}1-{\frac {x^{2}}{c^{2}}}{\Big )}{\bigg \}},$

for 0 ≤ x < c. Here χ, and c are parameters of the distribution and

$\Psi (\chi )=\Phi (\chi )-\chi \phi (\chi )-{\tfrac {1}{2}},$

and Φ(·), ϕ(·) are the cumulative distribution and probability density functions of the standard normal distribution, respectively.

Cumulative distribution function

The cdf of the ARGUS distribution is

$F(x)=1-{\frac {\Psi {\Big (}\chi {\sqrt {1-x^{2}/c^{2}}}\,{\Big )}}{\Psi (\chi )}}.$

Parameter estimation

Parameter c is assumed to be known (the speed of light), whereas χ can be estimated from the sample X₁, …, 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:

$f(x)={\frac {2^{{-p}}\chi ^{{2(p+1)}}}{\Gamma (p+1)-\Gamma (p+1,\,{\tfrac {1}{2}}\chi ^{2})}}\ \cdot \ {\frac {x}{c^{2}}}{\bigg (}1-{\frac {x^{2}}{c^{2}}}{\bigg )}^{p}\exp {\bigg \{}-{\frac 12}\chi ^{2}{\Big (}1-{\frac {x^{2}}{c^{2}}}{\Big )}{\bigg \}},\qquad 0\leq x\leq c,$

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.

mode = ${\frac {c}{{\sqrt 2}\chi }}{\sqrt {(\chi ^{2}-2p-1)+{\sqrt {\chi ^{2}(\chi ^{2}-4p+2)+(1+2p)^{2}}}}}$

p = 0.5 gives a regular ARGUS, listed above.

References

↑ Albrecht, H. (1990). "Search for hadronic b→u decays". Physics Letters B 241 (2): 278–282. 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.

Albrecht, H. (1994). "Measurement of the polarization in the decay B → J/ψK*". Physics Letters B 340 (3): 217–220. doi:10.1016/0370-2693(94)01302-0.
Pedlar, T.; Cronin-Hennessy, D.; Hietala, J.; Dobbs, S.; Metreveli, Z.; Seth, K.; Tomaradze, A.; Xiao, T.; Martin, L. (2011). "Observation of the h_{c}(1P) Using e^{+}e^{-} Collisions above the DD¯ Threshold". Physical Review Letters 107 (4). arXiv:1104.2025. Bibcode:2011PhRvL.107d1803P. doi:10.1103/PhysRevLett.107.041803.
Lees, J. P.; Poireau, V.; Prencipe, E.; Tisserand, V.; Garra Tico, J.; Grauges, E.; Martinelli, M.; Palano, A.; Pappagallo, M.; Eigen, G.; Stugu, B.; Sun, L.; Battaglia, M.; Brown, D. N.; Hooberman, B.; Kerth, L. T.; Kolomensky, Y. G.; Lynch, G.; Osipenkov, I. L.; Tanabe, T.; Hawkes, C. M.; Soni, N.; Watson, A. T.; Koch, H.; Schroeder, T.; Asgeirsson, D. J.; Hearty, C.; Mattison, T. S.; McKenna, J. A.; Barrett, M. (2010). "Search for Charged Lepton Flavor Violation in Narrow Υ Decays". Physical Review Letters 104 (15). arXiv:1001.1883. Bibcode:2010PhRvL.104o1802L. doi:10.1103/PhysRevLett.104.151802.

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support

beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Triangular U-quadratic uniform Wigner semicircle Xenakis

[[List of probability distributions#Supported_on_semi-infinite_intervals.2C_usually_.5B0.2C.E2.88.9E.29|Continuous univariate supported on a semi-infinite interval, usually [0,∞)]]

Benini
Benktander 1st kind
Benktander 2nd kind
Beta prime
Burr
chi-squared
chi
Coxian
Dagum
Davis
EL
Erlang
exponential
F
folded normal
Flory-Schulz
Fréchet
Gamma
Gamma/Gompertz
generalized inverse Gaussian
Gompertz
half-logistic
half-normal
Hotelling's T-squared
hyper-Erlang
hyperexponential
hypoexponential
inverse chi-squared (scaled inverse chi-squared)
inverse Gaussian
inverse gamma
Kolmogorov
Lévy
log-Cauchy
log-Laplace
log-logistic
log-normal
Maxwell–Boltzmann
Maxwell–Jüttner
Mittag–Leffler
Nakagami
noncentral chi-squared
Pareto
phase-type
Poly-Weibull
Rayleigh
relativistic Breit–Wigner
Rice
Rosin–Rammler
shifted Gompertz
truncated normal
type-2 Gumbel
Weibull
Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson SU Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

v t e Some common univariate probability distributions

Continuous	beta Cauchy chi-squared exponential F gamma Laplace log-normal normal Pareto Student's t uniform Weibull

Discrete	Bernoulli binomial discrete uniform geometric hypergeometric negative binomial Poisson

List of probability distributions

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