- Source: 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}
, and hence named after, by Jess Marcum for pulsed radars.
Properties
= Finite integral representation
=Using the fact that
Q
ν
(
a
,
0
)
=
1
{\displaystyle Q_{\nu }(a,0)=1}
, the generalized Marcum Q-function can alternatively be defined as a finite integral as
Q
ν
(
a
,
b
)
=
1
−
1
a
ν
−
1
∫
0
b
x
ν
exp
(
−
x
2
+
a
2
2
)
I
ν
−
1
(
a
x
)
d
x
.
{\displaystyle 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
ν
=
n
{\displaystyle \nu =n}
, such a representation is given by the trigonometric integral
Q
n
(
a
,
b
)
=
{
H
n
(
a
,
b
)
a
<
b
,
1
2
+
H
n
(
a
,
a
)
a
=
b
,
1
+
H
n
(
a
,
b
)
a
>
b
,
{\displaystyle Q_{n}(a,b)=\left\{{\begin{array}{lr}H_{n}(a,b)&a
where
H
n
(
a
,
b
)
=
ζ
1
−
n
2
π
exp
(
−
a
2
+
b
2
2
)
∫
0
2
π
cos
(
n
−
1
)
θ
−
ζ
cos
n
θ
1
−
2
ζ
cos
θ
+
ζ
2
exp
(
a
b
cos
θ
)
d
θ
,
{\displaystyle 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
ζ
=
a
/
b
{\displaystyle \zeta =a/b}
is a constant.
For any real
ν
>
0
{\displaystyle \nu >0}
, such finite trigonometric integral is given by
Q
ν
(
a
,
b
)
=
{
H
ν
(
a
,
b
)
+
C
ν
(
a
,
b
)
a
<
b
,
1
2
+
H
ν
(
a
,
a
)
+
C
ν
(
a
,
b
)
a
=
b
,
1
+
H
ν
(
a
,
b
)
+
C
ν
(
a
,
b
)
a
>
b
,
{\displaystyle Q_{\nu }(a,b)=\left\{{\begin{array}{lr}H_{\nu }(a,b)+C_{\nu }(a,b)&a
where
H
n
(
a
,
b
)
{\displaystyle H_{n}(a,b)}
is as defined before,
ζ
=
a
/
b
{\displaystyle \zeta =a/b}
, and the additional correction term is given by
C
ν
(
a
,
b
)
=
sin
(
ν
π
)
π
exp
(
−
a
2
+
b
2
2
)
∫
0
1
(
x
/
ζ
)
ν
−
1
ζ
+
x
exp
[
−
a
b
2
(
x
+
1
x
)
]
d
x
.
{\displaystyle 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
ν
{\displaystyle \nu }
, the correction term
C
ν
(
a
,
b
)
{\displaystyle C_{\nu }(a,b)}
tend to vanish.
= Monotonicity and log-concavity
=The generalized Marcum Q-function
Q
ν
(
a
,
b
)
{\displaystyle Q_{\nu }(a,b)}
is strictly increasing in
ν
{\displaystyle \nu }
and
a
{\displaystyle a}
for all
a
≥
0
{\displaystyle a\geq 0}
and
b
,
ν
>
0
{\displaystyle b,\nu >0}
, and is strictly decreasing in
b
{\displaystyle b}
for all
a
,
b
≥
0
{\displaystyle a,b\geq 0}
and
ν
>
0.
{\displaystyle \nu >0.}
The function
ν
↦
Q
ν
(
a
,
b
)
{\displaystyle \nu \mapsto Q_{\nu }(a,b)}
is log-concave on
[
1
,
∞
)
{\displaystyle [1,\infty )}
for all
a
,
b
≥
0.
{\displaystyle a,b\geq 0.}
The function
b
↦
Q
ν
(
a
,
b
)
{\displaystyle b\mapsto Q_{\nu }(a,b)}
is strictly log-concave on
(
0
,
∞
)
{\displaystyle (0,\infty )}
for all
a
≥
0
{\displaystyle a\geq 0}
and
ν
>
1
{\displaystyle \nu >1}
, which implies that the generalized Marcum Q-function satisfies the new-is-better-than-used property.
The function
a
↦
1
−
Q
ν
(
a
,
b
)
{\displaystyle a\mapsto 1-Q_{\nu }(a,b)}
is log-concave on
[
0
,
∞
)
{\displaystyle [0,\infty )}
for all
b
,
ν
>
0.
{\displaystyle b,\nu >0.}
= Series representation
=The generalized Marcum Q function of order
ν
>
0
{\displaystyle \nu >0}
can be represented using incomplete Gamma function as
Q
ν
(
a
,
b
)
=
1
−
e
−
a
2
/
2
∑
k
=
0
∞
1
k
!
γ
(
ν
+
k
,
b
2
2
)
Γ
(
ν
+
k
)
(
a
2
2
)
k
,
{\displaystyle 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
γ
(
s
,
x
)
{\displaystyle \gamma (s,x)}
is the lower incomplete Gamma function. This is usually called the canonical representation of the
ν
{\displaystyle \nu }
-th order generalized Marcum Q-function.
The generalized Marcum Q function of order
ν
>
0
{\displaystyle \nu >0}
can also be represented using generalized Laguerre polynomials as
Q
ν
(
a
,
b
)
=
1
−
e
−
a
2
/
2
∑
k
=
0
∞
(
−
1
)
k
L
k
(
ν
−
1
)
(
a
2
2
)
Γ
(
ν
+
k
+
1
)
(
b
2
2
)
k
+
ν
,
{\displaystyle 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
(
α
)
(
⋅
)
{\displaystyle L_{k}^{(\alpha )}(\cdot )}
is the generalized Laguerre polynomial of degree
k
{\displaystyle k}
and of order
α
{\displaystyle \alpha }
.
The generalized Marcum Q-function of order
ν
>
0
{\displaystyle \nu >0}
can also be represented as Neumann series expansions
Q
ν
(
a
,
b
)
=
e
−
(
a
2
+
b
2
)
/
2
∑
α
=
1
−
ν
∞
(
a
b
)
α
I
−
α
(
a
b
)
,
{\displaystyle 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
ν
(
a
,
b
)
=
e
−
(
a
2
+
b
2
)
/
2
∑
α
=
ν
∞
(
b
a
)
α
I
α
(
a
b
)
,
{\displaystyle 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
α
{\displaystyle \alpha }
assumes an integer value, we have
I
α
(
a
b
)
=
I
−
α
(
a
b
)
{\displaystyle I_{\alpha }(ab)=I_{-\alpha }(ab)}
.
For non-negative half-integer values
ν
=
n
+
1
/
2
{\displaystyle \nu =n+1/2}
, we have a closed form expression for the generalized Marcum Q-function as
Q
n
+
1
/
2
(
a
,
b
)
=
1
2
[
e
r
f
c
(
b
−
a
2
)
+
e
r
f
c
(
b
+
a
2
)
]
+
e
−
(
a
2
+
b
2
)
/
2
∑
k
=
1
n
(
b
a
)
k
−
1
/
2
I
k
−
1
/
2
(
a
b
)
,
{\displaystyle 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
e
r
f
c
(
⋅
)
{\displaystyle \mathrm {erfc} (\cdot )}
is the complementary error function. Since Bessel functions with half-integer parameter have finite sum expansions as
I
±
(
n
+
0.5
)
(
z
)
=
1
π
∑
k
=
0
n
(
n
+
k
)
!
k
!
(
n
−
k
)
!
[
(
−
1
)
k
e
z
∓
(
−
1
)
n
e
−
z
(
2
z
)
k
+
0.5
]
,
{\displaystyle 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
{\displaystyle 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
)
+
1
b
2
π
∑
i
=
1
n
(
b
a
)
i
∑
k
=
0
i
−
1
(
i
+
k
−
1
)
!
k
!
(
i
−
k
−
1
)
!
[
(
−
1
)
k
e
−
(
a
−
b
)
2
/
2
+
(
−
1
)
i
e
−
(
a
+
b
)
2
/
2
(
2
a
b
)
k
]
,
{\displaystyle 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
{\displaystyle n}
, where
Q
(
⋅
)
{\displaystyle 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
+
1
2
(
z
)
=
2
z
π
[
g
n
(
z
)
sinh
(
z
)
+
g
−
n
−
1
(
z
)
cosh
(
z
)
]
,
{\displaystyle 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
{\displaystyle g_{0}(z)=z^{-1}}
,
g
1
(
z
)
=
−
z
−
2
{\displaystyle g_{1}(z)=-z^{-2}}
, and
g
n
−
1
(
z
)
−
g
n
+
1
(
z
)
=
(
2
n
+
1
)
z
−
1
g
n
(
z
)
{\displaystyle g_{n-1}(z)-g_{n+1}(z)=(2n+1)z^{-1}g_{n}(z)}
for any integer value of
n
{\displaystyle n}
.
= Recurrence relation and generating function
=Integrating by parts, we can show that generalized Marcum Q-function satisfies the following recurrence relation
Q
ν
+
1
(
a
,
b
)
−
Q
ν
(
a
,
b
)
=
(
b
a
)
ν
e
−
(
a
2
+
b
2
)
/
2
I
ν
(
a
b
)
.
{\displaystyle 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
ν
−
n
(
a
,
b
)
=
Q
ν
(
a
,
b
)
−
(
b
a
)
ν
e
−
(
a
2
+
b
2
)
/
2
∑
k
=
1
n
(
a
b
)
k
I
ν
−
k
(
a
b
)
,
{\displaystyle 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
ν
+
n
(
a
,
b
)
=
Q
ν
(
a
,
b
)
+
(
b
a
)
ν
e
−
(
a
2
+
b
2
)
/
2
∑
k
=
0
n
−
1
(
b
a
)
k
I
ν
+
k
(
a
b
)
,
{\displaystyle 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
{\displaystyle n}
. The former recurrence can be used to formally define the generalized Marcum Q-function for negative
ν
{\displaystyle \nu }
. Taking
Q
∞
(
a
,
b
)
=
1
{\displaystyle Q_{\infty }(a,b)=1}
and
Q
−
∞
(
a
,
b
)
=
0
{\displaystyle Q_{-\infty }(a,b)=0}
for
n
=
∞
{\displaystyle n=\infty }
, we obtain the Neumann series representation of the generalized Marcum Q-function.
The related three-term recurrence relation is given by
Q
ν
+
1
(
a
,
b
)
−
(
1
+
c
ν
(
a
,
b
)
)
Q
ν
(
a
,
b
)
+
c
ν
(
a
,
b
)
Q
ν
−
1
(
a
,
b
)
=
0
,
{\displaystyle 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
ν
(
a
,
b
)
=
(
b
a
)
I
ν
(
a
b
)
I
ν
+
1
(
a
b
)
.
{\displaystyle 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
a
2
2
Q
ν
+
2
(
a
,
b
)
=
(
a
2
2
−
ν
)
Q
ν
+
1
(
a
,
b
)
+
(
b
2
2
+
ν
)
Q
ν
(
a
,
b
)
−
b
2
2
Q
ν
−
1
(
a
,
b
)
.
{\displaystyle {\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
ν
+
1
(
a
,
b
)
=
Q
ν
(
a
,
b
)
+
1
a
∂
∂
a
Q
ν
(
a
,
b
)
,
{\displaystyle Q_{\nu +1}(a,b)=Q_{\nu }(a,b)+{\frac {1}{a}}{\frac {\partial }{\partial a}}Q_{\nu }(a,b),}
Q
ν
−
1
(
a
,
b
)
=
Q
ν
(
a
,
b
)
+
1
b
∂
∂
b
Q
ν
(
a
,
b
)
.
{\displaystyle 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
ν
(
a
,
b
)
{\displaystyle Q_{\nu }(a,b)}
for integral
ν
{\displaystyle \nu }
is
∑
n
=
−
∞
∞
t
n
Q
n
(
a
,
b
)
=
e
−
(
a
2
+
b
2
)
/
2
t
1
−
t
e
(
b
2
t
+
a
2
/
t
)
/
2
,
{\displaystyle \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.
{\displaystyle |t|<1.}
= Symmetry relation
=Using the two Neumann series representations, we can obtain the following symmetry relation for positive integral
ν
=
n
{\displaystyle \nu =n}
Q
n
(
a
,
b
)
+
Q
n
(
b
,
a
)
=
1
+
e
−
(
a
2
+
b
2
)
/
2
[
I
0
(
a
b
)
+
∑
k
=
1
n
−
1
a
2
k
+
b
2
k
(
a
b
)
k
I
k
(
a
b
)
]
.
{\displaystyle 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
{\displaystyle n=1}
we have
Q
1
(
a
,
b
)
+
Q
1
(
b
,
a
)
=
1
+
e
−
(
a
2
+
b
2
)
/
2
I
0
(
a
b
)
.
{\displaystyle 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
ν
(
0
,
0
)
=
1
,
{\displaystyle Q_{\nu }(0,0)=1,}
Q
ν
(
a
,
0
)
=
1
,
{\displaystyle Q_{\nu }(a,0)=1,}
Q
ν
(
a
,
+
∞
)
=
0
,
{\displaystyle Q_{\nu }(a,+\infty )=0,}
Q
ν
(
0
,
b
)
=
Γ
(
ν
,
b
2
/
2
)
Γ
(
ν
)
,
{\displaystyle Q_{\nu }(0,b)={\frac {\Gamma (\nu ,b^{2}/2)}{\Gamma (\nu )}},}
Q
ν
(
+
∞
,
b
)
=
1
,
{\displaystyle Q_{\nu }(+\infty ,b)=1,}
Q
∞
(
a
,
b
)
=
1
,
{\displaystyle Q_{\infty }(a,b)=1,}
For
a
=
b
{\displaystyle a=b}
, by subtracting the two forms of Neumann series representations, we have
Q
1
(
a
,
a
)
=
1
2
[
1
+
e
−
a
2
I
0
(
a
2
)
]
,
{\displaystyle 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
)
=
1
2
[
1
+
e
−
a
2
I
0
(
a
2
)
]
+
e
−
a
2
∑
k
=
1
n
−
1
I
k
(
a
2
)
,
{\displaystyle 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
)
=
1
2
[
1
+
e
−
a
2
I
0
(
a
2
)
]
−
e
−
a
2
∑
k
=
1
n
I
k
(
a
2
)
,
{\displaystyle 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
{\displaystyle n}
.
For
ν
=
1
/
2
{\displaystyle \nu =1/2}
, using the basic integral definition of generalized Marcum Q-function, we have
Q
1
/
2
(
a
,
b
)
=
1
2
[
e
r
f
c
(
b
−
a
2
)
+
e
r
f
c
(
b
+
a
2
)
]
.
{\displaystyle 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
ν
=
3
/
2
{\displaystyle \nu =3/2}
, we have
Q
3
/
2
(
a
,
b
)
=
Q
1
/
2
(
a
,
b
)
+
2
π
sinh
(
a
b
)
a
e
−
(
a
2
+
b
2
)
/
2
.
{\displaystyle 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
ν
=
5
/
2
{\displaystyle \nu =5/2}
we have
Q
5
/
2
(
a
,
b
)
=
Q
3
/
2
(
a
,
b
)
+
2
π
a
b
cosh
(
a
b
)
−
sinh
(
a
b
)
a
3
e
−
(
a
2
+
b
2
)
/
2
.
{\displaystyle 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
ν
{\displaystyle \nu }
to be fixed and
a
b
{\displaystyle ab}
large, let
ζ
=
a
/
b
>
0
{\displaystyle \zeta =a/b>0}
, then the generalized Marcum-Q function has the following asymptotic form
Q
ν
(
a
,
b
)
∼
∑
n
=
0
∞
ψ
n
,
{\displaystyle Q_{\nu }(a,b)\sim \sum _{n=0}^{\infty }\psi _{n},}
where
ψ
n
{\displaystyle \psi _{n}}
is given by
ψ
n
=
1
2
ζ
ν
2
π
(
−
1
)
n
[
A
n
(
ν
−
1
)
−
ζ
A
n
(
ν
)
]
ϕ
n
.
{\displaystyle \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
ϕ
n
{\displaystyle \phi _{n}}
and
A
n
{\displaystyle A_{n}}
are given by
ϕ
n
=
[
(
b
−
a
)
2
2
a
b
]
n
−
1
2
Γ
(
1
2
−
n
,
(
b
−
a
)
2
2
)
,
{\displaystyle \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
(
ν
)
=
2
−
n
Γ
(
1
2
+
ν
+
n
)
n
!
Γ
(
1
2
+
ν
−
n
)
.
{\displaystyle A_{n}(\nu )={\frac {2^{-n}\Gamma ({\frac {1}{2}}+\nu +n)}{n!\Gamma ({\frac {1}{2}}+\nu -n)}}.}
The function
A
n
(
ν
)
{\displaystyle A_{n}(\nu )}
satisfies the recursion
A
n
+
1
(
ν
)
=
−
(
2
n
+
1
)
2
−
4
ν
2
8
(
n
+
1
)
A
n
(
ν
)
,
{\displaystyle A_{n+1}(\nu )=-{\frac {(2n+1)^{2}-4\nu ^{2}}{8(n+1)}}A_{n}(\nu ),}
for
n
≥
0
{\displaystyle n\geq 0}
and
A
0
(
ν
)
=
1.
{\displaystyle A_{0}(\nu )=1.}
In the first term of the above asymptotic approximation, we have
ϕ
0
=
2
π
a
b
b
−
a
e
r
f
c
(
b
−
a
2
)
.
{\displaystyle \phi _{0}={\frac {\sqrt {2\pi ab}}{b-a}}\mathrm {erfc} \left({\frac {b-a}{\sqrt {2}}}\right).}
Hence, assuming
b
>
a
{\displaystyle b>a}
, the first term asymptotic approximation of the generalized Marcum-Q function is
Q
ν
(
a
,
b
)
∼
ψ
0
=
(
b
a
)
ν
−
1
2
Q
(
b
−
a
)
,
{\displaystyle Q_{\nu }(a,b)\sim \psi _{0}=\left({\frac {b}{a}}\right)^{\nu -{\frac {1}{2}}}Q(b-a),}
where
Q
(
⋅
)
{\displaystyle Q(\cdot )}
is the Gaussian Q-function. Here
Q
ν
(
a
,
b
)
∼
0.5
{\displaystyle Q_{\nu }(a,b)\sim 0.5}
as
a
↑
b
.
{\displaystyle a\uparrow b.}
For the case when
a
>
b
{\displaystyle a>b}
, we have
Q
ν
(
a
,
b
)
∼
1
−
ψ
0
=
1
−
(
b
a
)
ν
−
1
2
Q
(
a
−
b
)
.
{\displaystyle Q_{\nu }(a,b)\sim 1-\psi _{0}=1-\left({\frac {b}{a}}\right)^{\nu -{\frac {1}{2}}}Q(a-b).}
Here too
Q
ν
(
a
,
b
)
∼
0.5
{\displaystyle Q_{\nu }(a,b)\sim 0.5}
as
a
↓
b
.
{\displaystyle a\downarrow b.}
= Differentiation
=The partial derivative of
Q
ν
(
a
,
b
)
{\displaystyle Q_{\nu }(a,b)}
with respect to
a
{\displaystyle a}
and
b
{\displaystyle b}
is given by
∂
∂
a
Q
ν
(
a
,
b
)
=
a
[
Q
ν
+
1
(
a
,
b
)
−
Q
ν
(
a
,
b
)
]
=
a
(
b
a
)
ν
e
−
(
a
2
+
b
2
)
/
2
I
ν
(
a
b
)
,
{\displaystyle {\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),}
∂
∂
b
Q
ν
(
a
,
b
)
=
b
[
Q
ν
−
1
(
a
,
b
)
−
Q
ν
(
a
,
b
)
]
=
−
b
(
b
a
)
ν
−
1
e
−
(
a
2
+
b
2
)
/
2
I
ν
−
1
(
a
b
)
.
{\displaystyle {\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
1
a
∂
∂
a
Q
ν
(
a
,
b
)
+
1
b
∂
∂
b
Q
ν
+
1
(
a
,
b
)
=
0.
{\displaystyle {\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
ν
(
a
,
b
)
{\displaystyle Q_{\nu }(a,b)}
with respect to its arguments is given by
∂
n
∂
a
n
Q
ν
(
a
,
b
)
=
n
!
(
−
a
)
n
∑
k
=
0
[
n
/
2
]
(
−
2
a
2
)
−
k
k
!
(
n
−
2
k
)
!
∑
p
=
0
n
−
k
(
−
1
)
p
(
n
−
k
p
)
Q
ν
+
p
(
a
,
b
)
,
{\displaystyle {\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),}
∂
n
∂
b
n
Q
ν
(
a
,
b
)
=
n
!
a
1
−
ν
2
n
b
n
−
ν
+
1
e
−
(
a
2
+
b
2
)
/
2
∑
k
=
[
n
/
2
]
n
(
−
2
b
2
)
k
(
n
−
k
)
!
(
2
k
−
n
)
!
∑
p
=
0
k
−
1
(
k
−
1
p
)
(
−
a
b
)
p
I
ν
−
p
−
1
(
a
b
)
.
{\displaystyle {\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
ν
2
(
a
,
b
)
>
Q
ν
−
1
(
a
,
b
)
+
Q
ν
+
1
(
a
,
b
)
2
>
Q
ν
−
1
(
a
,
b
)
Q
ν
+
1
(
a
,
b
)
{\displaystyle Q_{\nu }^{2}(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
≥
b
>
0
{\displaystyle a\geq b>0}
and
ν
>
1
{\displaystyle \nu >1}
.
Bounds
= 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
ν
↦
Q
ν
(
a
,
b
)
{\displaystyle \nu \mapsto Q_{\nu }(a,b)}
and the fact that we have closed form expression for
Q
ν
(
a
,
b
)
{\displaystyle Q_{\nu }(a,b)}
when
ν
{\displaystyle \nu }
is half-integer valued.
Let
⌊
x
⌋
0.5
{\displaystyle \lfloor x\rfloor _{0.5}}
and
⌈
x
⌉
0.5
{\displaystyle \lceil x\rceil _{0.5}}
denote the pair of half-integer rounding operators that map a real
x
{\displaystyle x}
to its nearest left and right half-odd integer, respectively, according to the relations
⌊
x
⌋
0.5
=
⌊
x
−
0.5
⌋
+
0.5
{\displaystyle \lfloor x\rfloor _{0.5}=\lfloor x-0.5\rfloor +0.5}
⌈
x
⌉
0.5
=
⌈
x
+
0.5
⌉
−
0.5
{\displaystyle \lceil x\rceil _{0.5}=\lceil x+0.5\rceil -0.5}
where
⌊
x
⌋
{\displaystyle \lfloor x\rfloor }
and
⌈
x
⌉
{\displaystyle \lceil x\rceil }
denote the integer floor and ceiling functions.
The monotonicity of the function
ν
↦
Q
ν
(
a
,
b
)
{\displaystyle \nu \mapsto Q_{\nu }(a,b)}
for all
a
≥
0
{\displaystyle a\geq 0}
and
b
>
0
{\displaystyle b>0}
gives us the following simple bound
Q
⌊
ν
⌋
0.5
(
a
,
b
)
<
Q
ν
(
a
,
b
)
<
Q
⌈
ν
⌉
0.5
(
a
,
b
)
.
{\displaystyle Q_{\lfloor \nu \rfloor _{0.5}}(a,b)
However, the relative error of this bound does not tend to zero when
b
→
∞
{\displaystyle b\to \infty }
. For integral values of
ν
=
n
{\displaystyle \nu =n}
, this bound reduces to
Q
n
−
0.5
(
a
,
b
)
<
Q
n
(
a
,
b
)
<
Q
n
+
0.5
(
a
,
b
)
.
{\displaystyle Q_{n-0.5}(a,b)
A very good approximation of the generalized Marcum Q-function for integer valued
ν
=
n
{\displaystyle \nu =n}
is obtained by taking the arithmetic mean of the upper and lower bound
Q
n
(
a
,
b
)
≈
Q
n
−
0.5
(
a
,
b
)
+
Q
n
+
0.5
(
a
,
b
)
2
.
{\displaystyle 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
ν
↦
Q
ν
(
a
,
b
)
{\displaystyle \nu \mapsto Q_{\nu }(a,b)}
on
[
1
,
∞
)
{\displaystyle [1,\infty )}
as
Q
ν
1
(
a
,
b
)
ν
2
−
v
Q
ν
2
(
a
,
b
)
v
−
ν
1
<
Q
ν
(
a
,
b
)
<
Q
ν
2
(
a
,
b
)
ν
2
−
ν
+
1
Q
ν
2
+
1
(
a
,
b
)
ν
2
−
ν
,
{\displaystyle Q_{\nu _{1}}(a,b)^{\nu _{2}-v}Q_{\nu _{2}}(a,b)^{v-\nu _{1}}
where
ν
1
=
⌊
ν
⌋
0.5
{\displaystyle \nu _{1}=\lfloor \nu \rfloor _{0.5}}
and
ν
2
=
⌈
ν
⌉
0.5
{\displaystyle \nu _{2}=\lceil \nu \rceil _{0.5}}
for
ν
≥
1.5
{\displaystyle \nu \geq 1.5}
. The tightness of this bound improves as either
a
{\displaystyle a}
or
ν
{\displaystyle \nu }
increases. The relative error of this bound converges to 0 as
b
→
∞
{\displaystyle b\to \infty }
. For integral values of
ν
=
n
{\displaystyle \nu =n}
, this bound reduces to
Q
n
−
0.5
(
a
,
b
)
Q
n
+
0.5
(
a
,
b
)
<
Q
n
(
a
,
b
)
<
Q
n
+
0.5
(
a
,
b
)
Q
n
+
0.5
(
a
,
b
)
Q
n
+
1.5
(
a
,
b
)
.
{\displaystyle {\sqrt {Q_{n-0.5}(a,b)Q_{n+0.5}(a,b)}}
= Cauchy-Schwarz bound
=Using the trigonometric integral representation for integer valued
ν
=
n
{\displaystyle \nu =n}
, the following Cauchy-Schwarz bound can be obtained
e
−
b
2
/
2
≤
Q
n
(
a
,
b
)
≤
exp
[
−
1
2
(
b
2
+
a
2
)
]
I
0
(
2
a
b
)
2
n
−
1
2
+
ζ
2
(
1
−
n
)
2
(
1
−
ζ
2
)
,
ζ
<
1
,
{\displaystyle 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
)
≤
exp
[
−
1
2
(
b
2
+
a
2
)
]
I
0
(
2
a
b
)
ζ
2
(
1
−
n
)
2
(
ζ
2
−
1
)
,
ζ
>
1
,
{\displaystyle 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
ζ
=
a
/
b
>
0
{\displaystyle \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
ζ
=
a
/
b
>
0
{\displaystyle \zeta =a/b>0}
, one such bound for integer valued
ν
=
n
{\displaystyle \nu =n}
is given as
e
−
(
b
+
a
)
2
/
2
≤
Q
n
(
a
,
b
)
≤
e
−
(
b
−
a
)
2
/
2
+
ζ
1
−
n
−
1
π
(
1
−
ζ
)
[
e
−
(
b
−
a
)
2
/
2
−
e
−
(
b
+
a
)
2
/
2
]
,
ζ
<
1
,
{\displaystyle 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
)
≥
1
−
1
2
[
e
−
(
a
−
b
)
2
/
2
−
e
−
(
a
+
b
)
2
/
2
]
,
ζ
>
1.
{\displaystyle 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
{\displaystyle n=1}
, the bound simplifies to give
e
−
(
b
+
a
)
2
/
2
≤
Q
1
(
a
,
b
)
≤
e
−
(
b
−
a
)
2
/
2
,
ζ
<
1
,
{\displaystyle e^{-(b+a)^{2}/2}\leq Q_{1}(a,b)\leq e^{-(b-a)^{2}/2},\qquad \zeta <1,}
1
−
1
2
[
e
−
(
a
−
b
)
2
/
2
−
e
−
(
a
+
b
)
2
/
2
]
≤
Q
1
(
a
,
b
)
,
ζ
>
1.
{\displaystyle 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
≤
Q
n
(
a
,
b
)
≤
1
2
2
n
−
1
2
+
ζ
2
(
1
−
n
)
2
(
1
−
ζ
2
)
[
e
−
(
b
−
a
)
2
/
2
+
e
−
(
b
+
a
)
2
/
2
]
,
ζ
<
1
{\displaystyle 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
)
≥
1
−
1
2
ζ
2
(
1
−
n
)
2
(
ζ
2
−
1
)
[
e
−
(
b
−
a
)
2
/
2
+
e
−
(
b
+
a
)
2
/
2
]
,
ζ
>
1.
{\displaystyle 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
ν
=
n
{\displaystyle \nu =n}
is an integer, is given by
(
1
−
2
λ
)
−
n
exp
(
−
λ
b
2
+
λ
n
a
2
1
−
2
λ
)
≥
{
Q
n
(
a
,
b
)
,
b
2
>
n
(
a
2
+
2
)
1
−
Q
n
(
a
,
b
)
,
b
2
<
n
(
a
2
+
2
)
{\displaystyle (1-2\lambda )^{-n}\exp \left(-\lambda b^{2}+{\frac {\lambda na^{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}
where the Chernoff parameter
(
0
<
λ
<
1
/
2
)
{\displaystyle (0<\lambda <1/2)}
has optimum value
λ
0
{\displaystyle \lambda _{0}}
of
λ
0
=
1
2
(
1
−
n
b
2
−
n
b
2
1
+
(
a
b
)
2
n
)
.
{\displaystyle \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
Q
1
(
a
,
b
)
=
{
1
,
i
f
b
<
c
1
−
β
0
e
−
1
2
(
a
2
+
(
β
0
)
2
)
I
0
(
a
β
0
)
(
b
−
β
0
)
+
Q
1
(
a
,
β
0
)
,
i
f
c
1
≤
b
≤
c
2
0
,
i
f
b
>
c
2
{\displaystyle {\begin{aligned}Q_{1}(a,b)={\begin{cases}1,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\mathrm {if} ~b
where
β
0
=
a
+
a
2
+
2
2
,
{\displaystyle {\begin{aligned}\beta _{0}={\frac {a+{\sqrt {a^{2}+2}}}{2}},\end{aligned}}}
c
1
(
a
)
=
max
(
0
,
β
0
+
Q
1
(
a
,
β
0
)
−
1
β
0
e
−
1
2
(
a
2
+
(
β
0
)
2
)
I
0
(
a
β
0
)
)
,
{\displaystyle {\begin{aligned}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{aligned}}}
and
c
2
(
a
)
=
β
0
+
Q
1
(
a
,
β
0
)
β
0
e
−
1
2
(
a
2
+
(
β
0
)
2
)
I
0
(
a
β
0
)
.
{\displaystyle {\begin{aligned}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{aligned}}}
Equivalent forms for efficient computation
It is convenient to re-express the Marcum Q-function as
P
N
(
X
,
Y
)
=
Q
N
(
2
N
X
,
2
Y
)
.
{\displaystyle P_{N}(X,Y)=Q_{N}({\sqrt {2NX}},{\sqrt {2Y}}).}
The
P
N
(
X
,
Y
)
{\displaystyle P_{N}(X,Y)}
can be interpreted as the detection probability of
N
{\displaystyle N}
incoherently integrated received signal samples of constant received signal-to-noise ratio,
X
{\displaystyle X}
, with a normalized detection threshold
Y
{\displaystyle Y}
. In this equivalent form of Marcum Q-function, for given
a
{\displaystyle a}
and
b
{\displaystyle b}
, we have
X
=
a
2
/
2
N
{\displaystyle X=a^{2}/2N}
and
Y
=
b
2
/
2
{\displaystyle Y=b^{2}/2}
. Many expressions exist that can represent
P
N
(
X
,
Y
)
{\displaystyle 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
)
=
∑
k
=
0
∞
e
−
N
X
(
N
X
)
k
k
!
∑
m
=
0
N
−
1
+
k
e
−
Y
Y
m
m
!
,
{\displaystyle 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
)
=
∑
m
=
0
N
−
1
e
−
Y
Y
m
m
!
+
∑
m
=
N
∞
e
−
Y
Y
m
m
!
(
1
−
∑
k
=
0
m
−
N
e
−
N
X
(
N
X
)
k
k
!
)
,
{\displaystyle 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
)
=
∑
m
=
N
∞
e
−
Y
Y
m
m
!
∑
k
=
0
m
−
N
e
−
N
X
(
N
X
)
k
k
!
,
{\displaystyle 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
)
=
∑
k
=
0
∞
e
−
N
X
(
N
X
)
k
k
!
(
1
−
∑
m
=
0
N
−
1
+
k
e
−
Y
Y
m
m
!
)
,
{\displaystyle 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
−
(
N
X
+
Y
)
∑
r
=
N
∞
(
Y
N
X
)
r
/
2
I
r
(
2
N
X
Y
)
.
{\displaystyle 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
∼
E
x
p
(
λ
)
{\displaystyle X\sim \mathrm {Exp} (\lambda )}
is a exponential distribution with rate parameter
λ
{\displaystyle \lambda }
, then its cdf is given by
F
X
(
x
)
=
1
−
Q
1
(
0
,
2
λ
x
)
{\displaystyle F_{X}(x)=1-Q_{1}\left(0,{\sqrt {2\lambda x}}\right)}
If
X
∼
E
r
l
a
n
g
(
k
,
λ
)
{\displaystyle X\sim \mathrm {Erlang} (k,\lambda )}
is a Erlang distribution with shape parameter
k
{\displaystyle k}
and rate parameter
λ
{\displaystyle \lambda }
, then its cdf is given by
F
X
(
x
)
=
1
−
Q
k
(
0
,
2
λ
x
)
{\displaystyle F_{X}(x)=1-Q_{k}\left(0,{\sqrt {2\lambda x}}\right)}
If
X
∼
χ
k
2
{\displaystyle X\sim \chi _{k}^{2}}
is a chi-squared distribution with
k
{\displaystyle k}
degrees of freedom, then its cdf is given by
F
X
(
x
)
=
1
−
Q
k
/
2
(
0
,
x
)
{\displaystyle F_{X}(x)=1-Q_{k/2}(0,{\sqrt {x}})}
If
X
∼
G
a
m
m
a
(
α
,
β
)
{\displaystyle X\sim \mathrm {Gamma} (\alpha ,\beta )}
is a gamma distribution with shape parameter
α
{\displaystyle \alpha }
and rate parameter
β
{\displaystyle \beta }
, then its cdf is given by
F
X
(
x
)
=
1
−
Q
α
(
0
,
2
β
x
)
{\displaystyle F_{X}(x)=1-Q_{\alpha }(0,{\sqrt {2\beta x}})}
If
X
∼
W
e
i
b
u
l
l
(
k
,
λ
)
{\displaystyle X\sim \mathrm {Weibull} (k,\lambda )}
is a Weibull distribution with shape parameters
k
{\displaystyle k}
and scale parameter
λ
{\displaystyle \lambda }
, then its cdf is given by
F
X
(
x
)
=
1
−
Q
1
(
0
,
2
(
x
λ
)
k
2
)
{\displaystyle F_{X}(x)=1-Q_{1}\left(0,{\sqrt {2}}\left({\frac {x}{\lambda }}\right)^{\frac {k}{2}}\right)}
If
X
∼
G
G
(
a
,
d
,
p
)
{\displaystyle X\sim \mathrm {GG} (a,d,p)}
is a generalized gamma distribution with parameters
a
,
d
,
p
{\displaystyle a,d,p}
, then its cdf is given by
F
X
(
x
)
=
1
−
Q
d
p
(
0
,
2
(
x
a
)
p
2
)
{\displaystyle F_{X}(x)=1-Q_{\frac {d}{p}}\left(0,{\sqrt {2}}\left({\frac {x}{a}}\right)^{\frac {p}{2}}\right)}
If
X
∼
χ
k
2
(
λ
)
{\displaystyle X\sim \chi _{k}^{2}(\lambda )}
is a non-central chi-squared distribution with non-centrality parameter
λ
{\displaystyle \lambda }
and
k
{\displaystyle k}
degrees of freedom, then its cdf is given by
F
X
(
x
)
=
1
−
Q
k
/
2
(
λ
,
x
)
{\displaystyle F_{X}(x)=1-Q_{k/2}({\sqrt {\lambda }},{\sqrt {x}})}
If
X
∼
R
a
y
l
e
i
g
h
(
σ
)
{\displaystyle X\sim \mathrm {Rayleigh} (\sigma )}
is a Rayleigh distribution with parameter
σ
{\displaystyle \sigma }
, then its cdf is given by
F
X
(
x
)
=
1
−
Q
1
(
0
,
x
σ
)
{\displaystyle F_{X}(x)=1-Q_{1}\left(0,{\frac {x}{\sigma }}\right)}
If
X
∼
M
a
x
w
e
l
l
(
σ
)
{\displaystyle X\sim \mathrm {Maxwell} (\sigma )}
is a Maxwell–Boltzmann distribution with parameter
σ
{\displaystyle \sigma }
, then its cdf is given by
F
X
(
x
)
=
1
−
Q
3
/
2
(
0
,
x
σ
)
{\displaystyle F_{X}(x)=1-Q_{3/2}\left(0,{\frac {x}{\sigma }}\right)}
If
X
∼
χ
k
{\displaystyle X\sim \chi _{k}}
is a chi distribution with
k
{\displaystyle k}
degrees of freedom, then its cdf is given by
F
X
(
x
)
=
1
−
Q
k
/
2
(
0
,
x
)
{\displaystyle F_{X}(x)=1-Q_{k/2}(0,x)}
If
X
∼
N
a
k
a
g
a
m
i
(
m
,
Ω
)
{\displaystyle X\sim \mathrm {Nakagami} (m,\Omega )}
is a Nakagami distribution with
m
{\displaystyle m}
as shape parameter and
Ω
{\displaystyle \Omega }
as spread parameter, then its cdf is given by
F
X
(
x
)
=
1
−
Q
m
(
0
,
2
m
Ω
x
)
{\displaystyle F_{X}(x)=1-Q_{m}\left(0,{\sqrt {\frac {2m}{\Omega }}}x\right)}
If
X
∼
R
i
c
e
(
ν
,
σ
)
{\displaystyle X\sim \mathrm {Rice} (\nu ,\sigma )}
is a Rice distribution with parameters
ν
{\displaystyle \nu }
and
σ
{\displaystyle \sigma }
, then its cdf is given by
F
X
(
x
)
=
1
−
Q
1
(
ν
σ
,
x
σ
)
{\displaystyle F_{X}(x)=1-Q_{1}\left({\frac {\nu }{\sigma }},{\frac {x}{\sigma }}\right)}
If
X
∼
χ
k
(
λ
)
{\displaystyle X\sim \chi _{k}(\lambda )}
is a non-central chi distribution with non-centrality parameter
λ
{\displaystyle \lambda }
and
k
{\displaystyle k}
degrees of freedom, then its cdf is given by
F
X
(
x
)
=
1
−
Q
k
/
2
(
λ
,
x
)
{\displaystyle F_{X}(x)=1-Q_{k/2}(\lambda ,x)}
Footnotes
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
Shnidman, David A. (1989): The Calculation of the Probability of Detection and the Generalized Marcum Q-Function, IEEE Transactions on Information Theory, 35(2), 389-400.
Weisstein, Eric W. Marcum Q-Function. From MathWorld—A Wolfram Web Resource. [1]
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