In statistics, the Q-function is the tail probability of the standard normal distribution
ϕ
(
x
)
. In other words, Q(x) is the probability that a normal (Gaussian) random variable will obtain a value larger than x standard deviations above the mean.
If the underlying random variable is y, then the proper argument to the tail probability is derived as:
x
=
y
−
μ
σ
which expresses the number of standard deviations away from the mean.
Other definitions of the Q-function, all of which are simple transformations of the normal cumulative distribution function, are also used occasionally.
Because of its relation to the cumulative distribution function of the normal distribution, the Q-function can also be expressed in terms of the error function, which is an important function in applied mathematics and physics.
Definition and basic properties
Formally, the Q-function is defined as
Q
(
x
)
=
1
2
π
∫
x
∞
exp
(
−
u
2
2
)
d
u
.
Thus,
Q
(
x
)
=
1
−
Q
(
−
x
)
=
1
−
Φ
(
x
)
,
where
Φ
(
x
)
is the cumulative distribution function of the normal Gaussian distribution.
The Q-function can be expressed in terms of the error function, or the complementary error function, as
Q
(
x
)
=
1
2
(
2
π
∫
x
/
2
∞
exp
(
−
t
2
)
d
t
)
=
1
2
−
1
2
erf
(
x
2
)
-or-
=
1
2
erfc
(
x
2
)
.
An alternative form of the Q-function known as Craig's formula, after its discoverer, is expressed as:
Q
(
x
)
=
1
π
∫
0
π
2
exp
(
−
x
2
2
sin
2
θ
)
d
θ
.
This expression is valid only for positive values of x, but it can be used in conjunction with Q(x) = 1 − Q(−x) to obtain Q(x) for negative values. This form is advantageous in that the range of integration is fixed and finite.
The Q-function is not an elementary function. However, the bounds
become increasingly tight for large
x, and are often useful.
Using the substitution
v =
u2/2, the upper bound is derived as follows:
Similarly, using
ϕ
′
(
u
)
=
−
u
ϕ
(
u
)
and the quotient rule,
Solving for
Q(
x) provides the lower bound.
The Chernoff bound of the Q-function is
Improved exponential bounds and a pure exponential approximation are
A tight approximation of
Q
(
x
)
for
x
∈
[
0
,
∞
)
is given by Karagiannidis & Lioumpas (2007) who showed for the appropriate choice of parameters
{
A
,
B
}
that
f
(
x
;
A
,
B
)
=
(
1
−
e
−
A
x
)
e
−
x
2
B
π
x
≈
erfc
(
x
)
.
The absolute error between
f
(
x
;
A
,
B
)
and
erfc
(
x
)
over the range
[
0
,
R
]
is minimized by evaluating
{
A
,
B
}
=
a
r
g
m
i
n
{
A
,
B
}
1
R
∫
0
R
|
f
(
x
;
A
,
B
)
−
erfc
(
x
)
|
d
x
.
Using
R
=
20
and numerically integrating, they found the minimum error occurred when
{
A
,
B
}
=
{
1.98
,
1.135
}
,
which gave a good approximation for
∀
x
≥
0.
Substituting these values and using the relationship between
Q
(
x
)
and
erfc
(
x
)
from above gives
Q
(
x
)
≈
(
1
−
e
−
1.4
x
)
e
−
x
2
2
1.135
2
π
x
,
x
≥
0.
Inverse Q
The inverse Q-function can be related to the inverse error functions:
Q
−
1
(
y
)
=
2
e
r
f
−
1
(
1
−
2
y
)
=
2
e
r
f
c
−
1
(
2
y
)
The function
Q
−
1
(
y
)
finds application in digital communications. It is usually expressed in dB and generally called Q-factor:
Q
-
f
a
c
t
o
r
=
20
log
10
(
Q
−
1
(
y
)
)
d
B
where y is the bit-error rate (BER) of the digitally modulated signal under analysis. For instance, for QPSK in additive white Gaussian noise, the Q-factor defined above coincides with the value in dB of the signal to noise ratio that yields a bit error rate equal to y.
The Q-function is well tabulated and can be computed directly in most of the mathematical software packages such as R and those available in Python, MATLAB and Mathematica. Some values of the Q-function are given below for reference.
The Q-function can be generalized to higher dimensions:
Q
(
x
)
=
P
(
X
≥
x
)
,
where
X
∼
N
(
0
,
Σ
)
follows the multivariate normal distribution with covariance
Σ
and the threshold is of the form
x
=
γ
Σ
l
∗
for some positive vector
l
∗
>
0
and positive constant
γ
>
0
. As in the one dimensional case, there is no simple analytical formula for the Q-function. Nevertheless, the Q-function can be approximated arbitrarily well as
γ
becomes larger and larger.
