Marcum Q-function

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In statistics, the Marcum-Q-function is defined as

is also defined as

with modified Bessel function 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]

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


  • 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]
  1. ^ A. Annamalai, C. Tellambura and John Matyjas (2009) A New Twist on the Generalized Marcum Q-Function Q_M(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
  2. ^ 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