Marcum Q-function

In statistics, the generalized Marcum Q-function of order ν {\displaystyle \nu } is defined as Q ν ( a , b ) = 1 a ν − 1 ∫ b ∞ x ν exp ⁡ ( − x 2 + a 2 2 ) I ν − 1 ( a x ) d x {\displaystyle Q_{\nu }(a,b)={\frac {1}{a^{\nu -1}}}\int _{b}^{\infty }x^{\nu }\exp \left(-{\frac {x^{2}+a^{2}}{2}}\right)I_{\nu -1}(ax)\,dx} where b ≥ 0 {\displaystyle b\geq 0} and a , ν > 0 {\displaystyle a,\nu >0} and I ν − 1 {\displaystyle I_{\nu -1}} is the modified Bessel function of first kind of order ν − 1 {\displaystyle \nu -1} . If b > 0 {\displaystyle b>0} , the integral converges for any ν {\displaystyle \nu } . The Marcum Q-function occurs as a complementary cumulative distribution function for noncentral chi, noncentral chi-squared, and Rice distributions. In engineering, this function appears in the study of radar systems, communication systems, queueing system, and signal processing. This function was first studied for ν = 1 {\displaystyle \nu =1} by, and hence named after, Jess Marcum for pulsed radars.