Marcum Q-function

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


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

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

Finite integral representation
Using the fact that $$Q_\nu (a,0) = 1$$, the generalized Marcum Q-function can alternatively be defined as a finite integral as


 * $$Q_\nu (a,b) = 1 - \frac{1}{a^{\nu-1}} \int_0^b x^\nu \exp \left( -\frac{x^2 + a^2}{2} \right) I_{\nu-1}(ax) \, dx. $$

However, it is preferable to have an integral representation of the Marcum Q-function such that (i) the limits of the integral are independent of the arguments of the function, (ii) and that the limits are finite, (iii) and that the integrand is a Gaussian function of these arguments. For positive integer values of $$\nu = n$$, such a representation is given by the trigonometric integral



Q_n(a,b) = \left\{ \begin{array}{lr} H_n(a,b) & a < b, \\ \frac{1}{2} + H_n(a,a) & a=b, \\ 1 + H_n(a,b) & a > b, \end{array} \right.$$

where


 * $$H_n(a,b) = \frac{\zeta^{1-n}}{2\pi} \exp\left(-\frac{a^2+b^2}{2}\right) \int_0^{2\pi} \frac{\cos(n-1)\theta - \zeta \cos n\theta}{1-2\zeta\cos\theta + \zeta^2} \exp(ab\cos\theta) \mathrm{d} \theta, $$

and the ratio $$\zeta = a/b$$ is a constant.

For any real $$\nu > 0$$, such finite trigonometric integral is given by



Q_\nu(a,b) = \left\{ \begin{array}{lr} H_\nu(a,b) + C_\nu(a,b) & a < b, \\ \frac{1}{2} + H_\nu(a,a) + C_\nu(a,b) & a=b, \\ 1 + H_\nu(a,b) + C_\nu(a,b) & a > b, \end{array} \right.$$

where $$H_n(a,b)$$ is as defined before, $$\zeta = a/b$$, and the additional correction term is given by


 * $$ C_\nu(a,b) = \frac{\sin(\nu\pi)}{\pi} \exp\left(-\frac{a^2+b^2}{2}\right) \int_0^1 \frac{(x/\zeta)^{\nu-1}}{\zeta+x} \exp\left[ -\frac{ab}{2}\left(x+\frac{1}{x}\right) \right] \mathrm{d}x. $$

For integer values of $$\nu$$, the correction term $$C_\nu(a,b)$$ tend to vanish.

Monotonicity and log-concavity

 * The generalized Marcum Q-function $$Q_\nu(a,b)$$ is strictly increasing in $$\nu$$ and $$a$$ for all $$a \geq 0$$ and $$b, \nu > 0$$, and is strictly decreasing in $$b$$ for all $$a, b \geq 0$$ and $$\nu>0.$$


 * The function $$\nu \mapsto Q_\nu(a,b)$$ is log-concave on $$[1,\infty)$$ for all $$a, b \geq 0.$$


 * The function $$b \mapsto Q_\nu(a,b)$$ is strictly log-concave on $$(0,\infty)$$ for all $$a \geq 0$$ and $$\nu > 1$$, which implies that the generalized Marcum Q-function satisfies the new-is-better-than-used property.


 * The function $$a \mapsto 1 - Q_\nu(a,b)$$ is log-concave on $$[0,\infty)$$ for all $$b, \nu > 0.$$

Series representation

 * The generalized Marcum Q function of order $$\nu > 0$$ can be represented using incomplete Gamma function as


 * $$ Q_\nu (a,b) = 1 - e^{-a^2/2} \sum_{k=0}^\infty \frac{1}{k!} \frac{\gamma(\nu+k,\frac{b^2}{2})}{\Gamma(\nu+k)} \left( \frac{a^2}{2} \right)^k,

$$


 * where $$\gamma(s,x)$$ is the lower incomplete Gamma function. This is usually called the canonical representation of the $$\nu$$-th order generalized Marcum Q-function.


 * The generalized Marcum Q function of order $$\nu > 0$$ can also be represented using generalized Laguerre polynomials as


 * $$ Q_{\nu}(a,b) = 1 - e^{-a^2/2} \sum_{k=0}^\infty (-1)^k \frac{L_k^{(\nu-1)}(\frac{a^2}{2})}{\Gamma(\nu+k+1)} \left(\frac{b^2}{2}\right)^{k+\nu}, $$


 * where $$L_k^{(\alpha)}(\cdot)$$ is the generalized Laguerre polynomial of degree $$k$$ and of order $$\alpha$$.


 * The generalized Marcum Q-function of order $$\nu > 0$$ can also be represented as Neumann series expansions


 * $$Q_\nu (a,b) = e^{-(a^2 + b^2)/2} \sum_{\alpha=1-\nu}^\infty \left( \frac{a}{b}\right)^\alpha I_{-\alpha}(ab),$$


 * $$1 - Q_\nu(a,b) = e^{-(a^2 + b^2)/2} \sum_{\alpha=\nu}^\infty \left( \frac{b}{a}\right)^\alpha I_{\alpha}(ab),$$


 * where the summations are in increments of one. Note that when $$\alpha$$ assumes an integer value, we have $$I_{\alpha}(ab) = I_{-\alpha}(ab)$$.


 * For non-negative half-integer values $$\nu = n + 1/2$$, we have a closed form expression for the generalized Marcum Q-function as


 * $$Q_{n+1/2}(a,b) = \frac{1}{2}\left[ \mathrm{erfc}\left(\frac{b-a}{\sqrt{2}}\right) + \mathrm{erfc}\left(\frac{b+a}{\sqrt{2}}\right) \right] + e^{-(a^2 + b^2)/2} \sum_{k=1}^n \left(\frac{b}{a}\right)^{k-1/2} I_{k-1/2}(ab), $$


 * where $$\mathrm{erfc}(\cdot)$$ is the complementary error function. Since Bessel functions with half-integer parameter have finite sum expansions as


 * $$I_{\pm(n+0.5)}(z) = \frac{1}{\sqrt{\pi}} \sum_{k=0}^n \frac{(n+k)!}{k!(n-k)!} \left[ \frac{(-1)^k e^z \mp (-1)^n e^{-z}}{(2z)^{k+0.5}} \right],$$


 * where $$n$$ is non-negative integer, we can exactly represent the generalized Marcum Q-function with half-integer parameter. More precisely, we have


 * $$Q_{n+1/2}(a,b) = Q(b-a) + Q(b+a) + \frac{1}{b\sqrt{2\pi}} \sum_{i=1}^{n} \left(\frac{b}{a}\right)^i \sum_{k=0}^{i-1} \frac{(i+k-1)!}{k!(i-k-1)!} \left[ \frac{(-1)^k e^{-(a-b)^2/2} + (-1)^i e^{-(a+b)^2/2}}{(2ab)^k} \right],$$


 * for non-negative integers $$n$$, where $$Q(\cdot)$$ is the Gaussian Q-function. Alternatively, we can also more compactly express the Bessel functions with half-integer as sum of hyperbolic sine and cosine functions:


 * $$I_{n+\frac{1}{2}}(z) = \sqrt{\frac{2z}{\pi}} \left[ g_n(z) \sinh(z) + g_{-n-1}(z) \cosh(z)\right], $$


 * where $$g_0(z) = z^{-1}$$, $$g_1(z) = -z^{-2}$$, and $$g_{n-1}(z) - g_{n+1}(z) = (2n+1) z^{-1} g_n(z)$$ for any integer value of $$n$$.

Recurrence relation and generating function

 * Integrating by parts, we can show that generalized Marcum Q-function satisfies the following recurrence relation


 * $$ Q_{\nu+1}(a,b) - Q_\nu(a,b) = \left( \frac{b}{a} \right)^{\nu} e^{-(a^2 + b^2)/2} I_{\nu}(ab). $$


 * The above formula is easily generalized as


 * $$Q_{\nu-n}(a,b) = Q_\nu(a,b) - \left(\frac{b}{a}\right)^\nu e^{-(a^2+b^2)/2}\sum_{k=1}^n \left(\frac{a}{b}\right)^k I_{\nu-k}(ab),$$


 * $$Q_{\nu+n}(a,b) = Q_\nu(a,b) + \left(\frac{b}{a}\right)^\nu e^{-(a^2+b^2)/2}\sum_{k=0}^{n-1} \left(\frac{b}{a}\right)^k I_{\nu+k}(ab),$$


 * for positive integer $$n$$. The former recurrence can be used to formally define the generalized Marcum Q-function for negative $$\nu$$. Taking $$Q_\infty(a,b)=1$$ and $$Q_{-\infty}(a,b)=0$$ for $$n = \infty$$, we obtain the Neumann series representation of the generalized Marcum Q-function.


 * The related three-term recurrence relation is given by


 * $$Q_{\nu+1}(a,b) - (1+c_\nu(a,b))Q_\nu(a,b) + c_\nu(a,b) Q_{\nu-1}(a,b) = 0,$$


 * where


 * $$c_\nu(a,b) = \left(\frac{b}{a}\right) \frac{I_\nu(ab)}{I_{\nu+1}(ab)}.$$


 * We can eliminate the occurrence of the Bessel function to give the third order recurrence relation


 * $$\frac{a^2}{2} Q_{\nu+2}(a,b) = \left(\frac{a^2}{2} - \nu\right) Q_{\nu+1}(a,b) + \left(\frac{b^2}{2} + \nu\right)Q_{\nu}(a,b) - \frac{b^2}{2} Q_{\nu-1}(a,b).$$


 * Another recurrence relationship, relating it with its derivatives, is given by


 * $$Q_{\nu+1}(a,b) = Q_\nu(a,b) + \frac{1}{a} \frac{\partial}{\partial a} Q_\nu(a,b),$$
 * $$Q_{\nu-1}(a,b) = Q_\nu(a,b) + \frac{1}{b} \frac{\partial}{\partial b} Q_\nu(a,b).$$


 * The ordinary generating function of $$Q_\nu(a,b)$$ for integral $$\nu$$ is


 * $$\sum_{n=-\infty}^\infty t^n Q_n(a,b) = e^{-(a^2+b^2)/2} \frac{t}{1-t} e^{(b^2 t + a^2/t)/2},$$


 * where $$|t|<1.$$

Symmetry relation

 * Using the two Neumann series representations, we can obtain the following symmetry relation for positive integral $$\nu = n$$


 * $$Q_n(a,b) + Q_n(b,a) = 1 + e^{-(a^2+b^2)/2} \left[ I_0(ab) + \sum_{k=1}^{n-1} \frac{a^{2k} + b^{2k}}{(ab)^k} I_k(ab) \right]. $$


 * In particular, for $$n = 1$$ we have


 * $$Q_1(a,b) + Q_1(b,a) = 1 + e^{-(a^2+b^2)/2} I_0(ab). $$

Special values
Some specific values of Marcum-Q function are
 * $$ Q_\nu(0,0) = 1, $$
 * $$ Q_\nu(a,0) = 1, $$
 * $$ Q_\nu(a,+\infty) = 0, $$
 * $$ Q_\nu(0,b) = \frac{\Gamma(\nu,b^2/2)}{\Gamma(\nu)}, $$
 * $$ Q_\nu(+\infty,b) = 1, $$
 * $$ Q_\infty(a,b) = 1, $$
 * For $$a=b$$, by subtracting the two forms of Neumann series representations, we have


 * $$Q_1(a,a) = \frac{1}{2}[1 + e^{-a^2}I_0(a^2)],$$


 * which when combined with the recursive formula gives


 * $$Q_n(a,a) = \frac{1}{2}[1 + e^{-a^2}I_0(a^2)] + e^{-a^2} \sum_{k=1}^{n-1} I_k(a^2),$$
 * $$Q_{-n}(a,a) = \frac{1}{2}[1 + e^{-a^2}I_0(a^2)] - e^{-a^2} \sum_{k=1}^{n} I_k(a^2),$$


 * for any non-negative integer $$n$$.


 * For $$\nu = 1/2$$, using the basic integral definition of generalized Marcum Q-function, we have


 * $$ Q_{1/2}(a,b) = \frac{1}{2}\left[ \mathrm{erfc}\left(\frac{b-a}{\sqrt{2}}\right) + \mathrm{erfc}\left(\frac{b+a}{\sqrt{2}}\right) \right].$$


 * For $$\nu=3/2$$, we have


 * $$Q_{3/2}(a,b) = Q_{1/2}(a,b) + \sqrt{\frac{2}{\pi}} \, \frac{\sinh(ab)}{a} e^{-(a^2 + b^2)/2}. $$


 * For $$\nu = 5/2$$ we have


 * $$Q_{5/2}(a,b) = Q_{3/2}(a,b) + \sqrt{\frac{2}{\pi}} \, \frac{ab \cosh (ab) - \sinh (ab) }{a^3} e^{-(a^2 + b^2)/2}.$$

Asymptotic forms

 * Assuming $$\nu$$ to be fixed and $$ab$$ large, let $$\zeta = a/b > 0$$, then the generalized Marcum-Q function has the following asymptotic form


 * $$Q_\nu(a,b) \sim \sum_{n=0}^\infty \psi_n,$$


 * where $$\psi_n$$ is given by


 * $$\psi_n = \frac{1}{2\zeta^\nu \sqrt{2\pi}} (-1)^n \left[ A_n(\nu-1) - \zeta A_n(\nu) \right] \phi_n.$$


 * The functions $$\phi_n$$ and $$A_n$$ are given by


 * $$\phi_n = \left[ \frac{(b-a)^2}{2ab} \right]^{n-\frac{1}{2}} \Gamma\left(\frac{1}{2} - n, \frac{(b-a)^2}{2}\right),$$


 * $$A_n(\nu) = \frac{2^{-n}\Gamma(\frac{1}{2}+\nu+n)}{n!\Gamma(\frac{1}{2}+\nu-n)}.$$


 * The function $$A_n(\nu)$$ satisfies the recursion


 * $$A_{n+1}(\nu) = - \frac{(2n+1)^2 - 4\nu^2}{8(n+1)}A_n(\nu),$$


 * for $$n \geq 0$$ and $$A_0(\nu)=1.$$


 * In the first term of the above asymptotic approximation, we have


 * $$\phi_0 = \frac{\sqrt{2 \pi ab}}{b-a} \mathrm{erfc}\left(\frac{b-a}{\sqrt{2}}\right).$$


 * Hence, assuming $$b>a$$, the first term asymptotic approximation of the generalized Marcum-Q function is


 * $$Q_\nu(a,b) \sim \psi_0 = \left(\frac{b}{a}\right)^{\nu-\frac{1}{2}} Q(b-a),$$


 * where $$Q(\cdot)$$ is the Gaussian Q-function. Here $$Q_\nu(a,b) \sim 0.5$$ as $$a \uparrow b.$$


 * For the case when $$a > b$$, we have


 * $$Q_\nu(a,b) \sim 1-\psi_0 = 1-\left(\frac{b}{a}\right)^{\nu-\frac{1}{2}} Q(a-b).$$


 * Here too $$Q_\nu(a,b) \sim 0.5$$ as $$a \downarrow b.$$

Differentiation

 * The partial derivative of $$Q_\nu(a,b)$$ with respect to $$a$$ and $$b$$ is given by


 * $$ \frac{\partial}{\partial a} Q_\nu(a,b) = a \left[Q_{\nu+1}(a,b) - Q_{\nu}(a,b)\right] = a \left(\frac{b}{a}\right)^{\nu} e^{-(a^2+b^2)/2} I_{\nu}(ab),$$
 * $$ \frac{\partial}{\partial b} Q_\nu(a,b) = b \left[Q_{\nu-1}(a,b) - Q_{\nu}(a,b)\right] = - b \left(\frac{b}{a}\right)^{\nu-1} e^{-(a^2+b^2)/2} I_{\nu-1}(ab).$$


 * We can relate the two partial derivatives as


 * $$\frac{1}{a}\frac{\partial}{\partial a} Q_\nu(a,b) + \frac{1}{b} \frac{\partial}{\partial b} Q_{\nu+1}(a,b) = 0.$$


 * The n-th partial derivative of $$Q_\nu(a,b)$$ with respect to its arguments is given by


 * $$ \frac{\partial^n}{\partial a^n} Q_\nu(a,b) = n! (-a)^n \sum_{k=0}^{[n/2]} \frac{(-2a^2)^{-k}}{k!(n-2k)!} \sum_{p=0}^{n-k} (-1)^p \binom{n-k}{p} Q_{\nu+p}(a,b), $$
 * $$ \frac{\partial^n}{\partial b^n} Q_\nu(a,b) = \frac{n! a^{1-\nu}}{2^n b^{n-\nu+1}} e^{-(a^2+b^2)/2} \sum_{k=[n/2]}^n \frac{(-2b^2)^k}{(n-k)!(2k-n)!} \sum_{p=0}^{k-1} \binom{k-1}{p} \left(-\frac{a}{b}\right)^p I_{\nu-p-1}(ab). $$

Inequalities

 * The generalized Marcum-Q function satisfies a Turán-type inequality


 * $$Q^2_\nu(a,b) > \frac{Q_{\nu-1}(a,b) + Q_{\nu+1}(a,b)}{2} > Q_{\nu-1}(a,b) Q_{\nu+1}(a,b)$$


 * for all $$a \geq b > 0$$ and $$\nu > 1$$.

Based on monotonicity and log-concavity
Various upper and lower bounds of generalized Marcum-Q function can be obtained using monotonicity and log-concavity of the function $$\nu \mapsto Q_\nu(a,b)$$ and the fact that we have closed form expression for $$Q_\nu(a,b)$$ when $$\nu$$ is half-integer valued.

Let $$\lfloor x \rfloor_{0.5}$$ and $$\lceil x \rceil_{0.5}$$ denote the pair of half-integer rounding operators that map a real $$x$$ to its nearest left and right half-odd integer, respectively, according to the relations


 * $$\lfloor x \rfloor_{0.5} = \lfloor x - 0.5 \rfloor + 0.5$$
 * $$ \lceil x \rceil_{0.5} = \lceil x + 0.5 \rceil - 0.5$$

where $$\lfloor x \rfloor$$ and $$\lceil x \rceil$$ denote the integer floor and ceiling functions.


 * The monotonicity of the function $$\nu \mapsto Q_\nu(a,b)$$ for all $$a \geq 0$$ and $$b > 0$$ gives us the following simple bound


 * $$Q_{\lfloor\nu\rfloor_{0.5}}(a,b) < Q_\nu(a,b) < Q_{\lceil\nu\rceil_{0.5}}(a,b).$$


 * However, the relative error of this bound does not tend to zero when $$b \to \infty$$. For integral values of $$\nu = n$$, this bound reduces to


 * $$Q_{n-0.5}(a,b) < Q_n(a,b) < Q_{n+0.5}(a,b).$$


 * A very good approximation of the generalized Marcum Q-function for integer valued $$\nu = n$$ is obtained by taking the arithmetic mean of the upper and lower bound


 * $$ Q_n(a,b) \approx \frac{Q_{n-0.5}(a,b) + Q_{n+0.5}(a,b)}{2}.$$


 * A tighter bound can be obtained by exploiting the log-concavity of $$\nu \mapsto Q_\nu(a,b)$$ on $$[1,\infty)$$ as


 * $$Q_{\nu_1}(a,b)^{\nu_2 - v} Q_{\nu_2}(a,b)^{v - \nu_1} < Q_\nu(a,b) < \frac{Q_{\nu_2}(a,b)^{\nu_2 - \nu + 1}}{Q_{\nu_2 + 1}(a,b)^{\nu_2 - \nu}},$$


 * where $$\nu_1 = \lfloor\nu\rfloor_{0.5}$$ and $$\nu_2 = \lceil\nu\rceil_{0.5}$$ for $$\nu \geq 1.5$$. The tightness of this bound improves as either $$a$$ or $$\nu$$ increases. The relative error of this bound converges to 0 as $$b \to \infty$$. For integral values of $$\nu = n$$, this bound reduces to


 * $$\sqrt{Q_{n - 0.5}(a,b) Q_{n + 0.5}(a,b)} < Q_n(a,b) < Q_{n + 0.5}(a,b) \sqrt{\frac{Q_{n + 0.5}(a,b)}{Q_{n + 1.5}(a,b)}}.$$

Cauchy-Schwarz bound
Using the trigonometric integral representation for integer valued $$\nu=n$$, the following Cauchy-Schwarz bound can be obtained


 * $$e^{-b^2/2} \leq Q_n(a,b) \leq \exp\left[-\frac{1}{2}(b^2 + a^2)\right] \sqrt{I_0(2ab)} \sqrt{\frac{2n-1}{2} + \frac{\zeta^{2(1-n)}}{2(1-\zeta^2)}}, \qquad \zeta < 1,$$
 * $$1 - Q_n(a,b) \leq \exp\left[-\frac{1}{2}(b^2+a^2)\right] \sqrt{I_0(2ab)} \sqrt{\frac{\zeta^{2(1-n)}}{2(\zeta^2-1)}}, \qquad \zeta > 1,$$

where $$\zeta = a/b >0$$.

Exponential-type bounds
For analytical purpose, it is often useful to have bounds in simple exponential form, even though they may not be the tightest bounds achievable. Letting $$\zeta = a/b >0$$, one such bound for integer valued $$\nu = n$$ is given as


 * $$e^{-(b+a)^2/2} \leq Q_n(a,b) \leq e^{-(b-a)^2/2} + \frac{\zeta^{1-n} - 1}{\pi(1-\zeta)} \left[e^{-(b-a)^2/2} - e^{-(b+a)^2/2} \right], \qquad \zeta < 1, $$
 * $$Q_n(a,b) \geq 1 - \frac{1}{2}\left[e^{-(a-b)^2/2} - e^{-(a+b)^2/2} \right], \qquad \zeta > 1.$$

When $$n=1$$, the bound simplifies to give


 * $$e^{-(b+a)^2/2} \leq Q_1(a,b) \leq e^{-(b-a)^2/2}, \qquad \zeta <1, $$
 * $$1 - \frac{1}{2}\left[e^{-(a-b)^2/2} - e^{-(a+b)^2/2} \right] \leq Q_1(a,b), \qquad \zeta > 1.$$

Another such bound obtained via Cauchy-Schwarz inequality is given as


 * $$e^{-b^2/2} \leq Q_n(a,b) \leq \frac{1}{2} \sqrt{\frac{2n-1}{2} + \frac{\zeta^{2(1-n)}}{2(1-\zeta^2)}} \left[ e^{-(b-a)^2/2} + e^{-(b+a)^2/2} \right], \qquad \zeta < 1$$
 * $$Q_n(a,b) \geq 1 - \frac{1}{2} \sqrt{\frac{\zeta^{2(1-n)}}{2(\zeta^2-1)}} \left[ e^{-(b-a)^2/2} + e^{-(b+a)^2/2} \right], \qquad \zeta > 1.$$

Chernoff-type bound
Chernoff-type bounds for the generalized Marcum Q-function, where $$\nu = n$$ is an integer, is given by


 * $$(1-2\lambda)^{-n} \exp \left(-\lambda b^2 + \frac{\lambda n a^2}{1 - 2\lambda} \right) \geq \left\{

\begin{array}{lr} Q_n(a,b), & b^2 > n(a^2+2) \\ 1 - Q_n(a,b), & b^2 < n(a^2+2) \end{array} \right.$$

where the Chernoff parameter $$(0 < \lambda < 1/2)$$ has optimum value $$\lambda_0$$ of


 * $$\lambda_0 = \frac{1}{2}\left(1 - \frac{n}{b^2} - \frac{n}{b^2} \sqrt{1 + \frac{(ab)^2}{n}}\right).$$

Semi-linear approximation
The first-order Marcum-Q function can be semi-linearly approximated by


 * $$\begin{align}

Q_1(a, b)= \begin{cases} 1, \mathrm{if}~ b < c_1 \\ -\beta_0 e^{-\frac{1}{2}\left(a^2+\left(\beta_0\right)^2\right)}I_0\left(a\beta_0\right)\left(b-\beta_0\right)+Q_1\left(a,\beta_0\right), \mathrm{if}~ c_1 \leq b \leq c_2 \\ 0, \mathrm{if}~ b> c_2 \end{cases} \end{align}$$ where

\begin{align} \beta_0 = \frac{a+\sqrt{a^2+2}}{2}, \end{align} $$

\begin{align} c_1(a) = \max\Bigg(0,\beta_0+\frac{Q_1\left(a,\beta_0\right)-1}{\beta_0 e^{-\frac{1}{2}\left(a^2+\left(\beta_0\right)^2\right)}I_0\left(a\beta_0\right)}\Bigg), \end{align} $$ and

\begin{align} c_2(a) = \beta_0+\frac{Q_1\left(a,\beta_0\right)}{\beta_0 e^{-\frac{1}{2}\left(a^2+\left(\beta_0\right)^2\right)}I_0\left(a\beta_0\right)}. \end{align} $$

Equivalent forms for efficient computation
It is convenient to re-express the Marcum Q-function as


 * $$ P_N(X,Y) = Q_N(\sqrt{2NX},\sqrt{2Y}). $$

The $$P_N(X,Y)$$ can be interpreted as the detection probability of $$N$$ incoherently integrated received signal samples of constant received signal-to-noise ratio, $$X$$, with a normalized detection threshold $$Y$$. In this equivalent form of Marcum Q-function, for given $$a$$ and $$b$$, we have $$X = a^2/2N$$ and $$Y = b^2/2$$. Many expressions exist that can represent $$P_N(X,Y)$$. However, the five most reliable, accurate, and efficient ones for numerical computation are given below. They are form one:


 * $$ P_N(X,Y) = \sum_{k=0}^\infty e^{-NX} \frac{(NX)^k}{k!} \sum_{m=0}^{N-1+k} e^{-Y} \frac{Y^m}{m!}, $$

form two:


 * $$ P_N(X,Y) = \sum_{m=0}^{N-1} e^{-Y} \frac{Y^m}{m!} + \sum_{m=N}^\infty e^{-Y} \frac{Y^m}{m!} \left( 1 - \sum_{k=0}^{m-N} e^{-NX} \frac{(NX)^k}{k!} \right), $$

form three:


 * $$ 1 - P_N(X,Y) = \sum_{m=N}^\infty e^{-Y} \frac{Y^m}{m!} \sum_{k=0}^{m-N} e^{-NX} \frac{(NX)^k}{k!}, $$

form four:


 * $$ 1 - P_N(X,Y) = \sum_{k=0}^\infty e^{-NX} \frac{(NX)^k}{k!} \left( 1 - \sum_{m=0}^{N-1+k} e^{-Y} \frac{Y^m}{m!} \right), $$

and form five:


 * $$ 1 - P_N(X,Y) = e^{-(NX+Y)} \sum_{r=N}^\infty \left(\frac{Y}{NX}\right)^{r/2} I_r(2\sqrt{NXY}). $$

Among these five form, the second form is the most robust.

Applications
The generalized Marcum Q-function can be used to represent the cumulative distribution function (cdf) of many random variables:


 * If $$X \sim \mathrm{Exp}(\lambda)$$ is a exponential distribution with rate parameter $$\lambda$$, then its cdf is given by $$F_X(x) = 1 - Q_1\left(0,\sqrt{2 \lambda x}\right)$$


 * If $$X \sim \mathrm{Erlang}(k,\lambda)$$ is a Erlang distribution with shape parameter $$k$$ and rate parameter $$\lambda$$, then its cdf is given by $$F_X(x) = 1 - Q_k\left(0,\sqrt{2 \lambda x}\right)$$


 * If $$X \sim \chi^2_k$$ is a chi-squared distribution with $$k$$ degrees of freedom, then its cdf is given by $$F_X(x) = 1 - Q_{k/2}(0,\sqrt{x})$$


 * If $$X \sim \mathrm{Gamma}(\alpha,\beta)$$ is a gamma distribution with shape parameter $$\alpha$$ and rate parameter $$\beta$$, then its cdf is given by $$F_X(x) = 1 - Q_{\alpha}(0,\sqrt{2 \beta x})$$


 * If $$X \sim \mathrm{Weibull}(k,\lambda)$$ is a Weibull distribution with shape parameters $$k$$ and scale parameter $$\lambda$$, then its cdf is given by $$F_X(x) = 1 - Q_1 \left( 0, \sqrt{2} \left(\frac{x}{\lambda}\right)^{\frac{k}{2}} \right)$$


 * If $$X \sim \mathrm{GG}(a,d,p)$$ is a generalized gamma distribution with parameters $$a, d, p$$, then its cdf is given by $$F_X(x) = 1 - Q_{\frac{d}{p}} \left( 0, \sqrt{2} \left(\frac{x}{a}\right)^{\frac{p}{2}} \right)$$
 * If $$X \sim \chi^2_k(\lambda)$$ is a non-central chi-squared distribution with non-centrality parameter $$\lambda$$ and $$k$$ degrees of freedom, then its cdf is given by $$F_X(x) = 1 - Q_{k/2}(\sqrt{\lambda},\sqrt{x})$$


 * If $$X \sim \mathrm{Rayleigh}(\sigma)$$ is a Rayleigh distribution with parameter $$\sigma$$, then its cdf is given by $$F_X(x) = 1 - Q_1\left(0,\frac{x}{\sigma}\right)$$


 * If $$X \sim \mathrm{Maxwell}(\sigma)$$ is a Maxwell–Boltzmann distribution with parameter $$\sigma$$, then its cdf is given by $$F_X(x) = 1 - Q_{3/2}\left(0,\frac{x}{\sigma}\right)$$


 * If $$X \sim \chi_k$$ is a chi distribution with $$k$$ degrees of freedom, then its cdf is given by $$F_X(x) = 1 - Q_{k/2}(0,x)$$


 * If $$X \sim \mathrm{Nakagami}(m,\Omega)$$ is a Nakagami distribution with $$m$$ as shape parameter and $$\Omega$$ as spread parameter, then its cdf is given by $$F_X(x) = 1 - Q_{m}\left(0,\sqrt{\frac{2m}{\Omega}}x\right)$$


 * If $$X \sim \mathrm{Rice}(\nu,\sigma)$$ is a Rice distribution with parameters $$\nu$$ and $$\sigma$$, then its cdf is given by $$F_X(x) = 1 - Q_1\left(\frac{\nu}{\sigma},\frac{x}{\sigma}\right)$$


 * If $$X \sim \chi_k(\lambda)$$ is a non-central chi distribution with non-centrality parameter $$\lambda$$ and $$k$$ degrees of freedom, then its cdf is given by $$F_X(x) = 1 - Q_{k/2}(\lambda,x)$$