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In statistics, the Marcum-Q-function \( Q_{M} \) is defined as

\( {\displaystyle Q_{M}(a,b)=\int _{b}^{\infty }x\left({\frac {x}{a}}\right)^{M-1}\exp \left(-{\frac {x^{2}+a^{2}}{2}}\right)I_{M-1}(ax)\,dx} \)

or as

\( {\displaystyle Q_{M}(a,b)=\exp \left(-{\frac {a^{2}+b^{2}}{2}}\right)\sum _{k=1-M}^{\infty }\left({\frac {a}{b}}\right)^{k}I_{k}(ab)} \)

with modified Bessel function \( {\displaystyle I_{M-1}} \) of order M − 1. The Marcum Q-function is used for example as a cumulative distribution function (more precisely, as a survivor function) for noncentral chi, noncentral chi-squared and Rice distributions.

For non-integer values of M, the Marcum Q function can be defined as[1]

\( {\displaystyle {\begin{aligned}Q_{M}(a,b)&=1-e^{-a^{2}/2}\sum _{k=0}^{\infty }\left({\frac {a^{2}}{2}}\right)^{k}{\frac {\gamma (M+k,{\frac {b^{2}}{2}})}{k!\Gamma (M+k)}}\\[6pt]&=1-e^{-a^{2}/2}\sum _{k=0}^{\infty }\left({\frac {a^{2}}{2}}\right)^{k}{\frac {P(M+k,{\frac {b^{2}}{2}})}{k!}}\end{aligned}}} \)

where \( P(s,x) \) is the regularized Gamma function.

The Marcum Q-function is monotonic and log-concave.[2]
References

Marcum, J. I. (1950) "Table of Q Functions". U.S. Air Force RAND Research Memorandum M-339. Santa Monica, CA: Rand Corporation, Jan. 1, 1950.
Nuttall, Albert H. (1975): Some Integrals Involving the QM Function, IEEE Transactions on Information Theory, 21(1), 95–96, ISSN 0018-9448
Weisstein, Eric W. Marcum Q-Function. From MathWorld—A Wolfram Web Resource. [1]

A. Annamalai, C. Tellambura and John Matyjas (2009) A New Twist on the Generalized Marcum Q-Function QM(a, b) with Fractional-Order M and Its Applications., 2009 6th IEEE Consumer Communications and Networking Conference, 1–5, ISBN 978-1-4244-2308-8
Yin Sun, Árpád Baricz, and Shidong Zhou (2010) On the Monotonicity, Log-Concavity, and Tight Bounds of the Generalized Marcum and Nuttall Q-Functions. IEEE Transactions on Information Theory, 56(3), 1166–1186, ISSN 0018-9448

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