In statistics, the Marcum-Q-function
Q
M
is defined as
Q
M
(
a
,
b
)
=
∫
b
∞
x
(
x
a
)
M
−
1
exp
(
−
x
2
+
a
2
2
)
I
M
−
1
(
a
x
)
d
x
Q
M
is also defined as
Q
M
(
a
,
b
)
=
exp
(
−
a
2
+
b
2
2
)
∑
k
=
1
−
M
∞
(
a
b
)
k
I
k
(
a
b
)
with modified Bessel function
I
M
−
1
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
Q
M
(
a
,
b
)
=
1
−
e
−
a
2
2
∑
k
=
0
∞
(
−
a
2
2
)
k
γ
(
M
+
k
,
b
2
2
)
k
!
Γ
(
M
+
k
)
The Marcum Q-function is monotonic and log-concave.
