Marcum Q-function

In statistics, the generalized Marcum Q-function of order ${\displaystyle \nu }$ is defined as

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

where ${\displaystyle b\geq 0}$ and ${\displaystyle a,\nu >0}$ and ${\displaystyle I_{\nu -1}}$ is the modified Bessel function of first kind of order ${\displaystyle \nu -1}$. If ${\displaystyle b>0}$, the integral converges for any ${\displaystyle \nu }$. 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 ${\displaystyle \nu =1}$, and hence named after, by Jess Marcum for pulsed radars.