A plot of the Q-function.
In statistics , the Q-function is the tail probability of the standard normal distribution
ϕ
(
x
)
{\displaystyle \phi (x)}
.[1] [2] 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
{\displaystyle x={\frac {y-\mu }{\sigma }}}
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.[3]
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
.
{\displaystyle Q(x)={\frac {1}{\sqrt {2\pi }}}\int _{x}^{\infty }\exp \left(-{\frac {u^{2}}{2}}\right)\,du.}
Thus,
Q
(
x
)
=
1
−
Q
(
−
x
)
=
1
−
Φ
(
x
)
,
{\displaystyle Q(x)=1-Q(-x)=1-\Phi (x)\,\!,}
where
Φ
(
x
)
{\displaystyle \Phi (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[2]
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
)
.
{\displaystyle {\begin{aligned}Q(x)&={\frac {1}{2}}\left({\frac {2}{\sqrt {\pi }}}\int _{x/{\sqrt {2}}}^{\infty }\exp \left(-t^{2}\right)\,dt\right)\\&={\frac {1}{2}}-{\frac {1}{2}}\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)~~{\text{ -or-}}\\&={\frac {1}{2}}\operatorname {erfc} \left({\frac {x}{\sqrt {2}}}\right).\end{aligned}}}
An alternative form of the Q -function known as Craig's formula, after its discoverer, is expressed as:[4]
Q
(
x
)
=
1
π
∫
0
π
2
exp
(
−
x
2
2
sin
2
θ
)
d
θ
.
{\displaystyle Q(x)={\frac {1}{\pi }}\int _{0}^{\frac {\pi }{2}}\exp \left(-{\frac {x^{2}}{2\sin ^{2}\theta }}\right)d\theta .}
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 finite.
(
x
1
+
x
2
)
ϕ
(
x
)
<
Q
(
x
)
<
ϕ
(
x
)
x
,
x
>
0
,
{\displaystyle \left({\frac {x}{1+x^{2}}}\right)\phi (x)<Q(x)<{\frac {\phi (x)}{x}},\qquad x>0,}
become increasingly tight for large x , and are often useful.
Using the substitution v =u 2 /2, the upper bound is derived as follows:
Q
(
x
)
=
∫
x
∞
ϕ
(
u
)
d
u
<
∫
x
∞
u
x
ϕ
(
u
)
d
u
=
∫
x
2
2
∞
e
−
v
x
2
π
d
v
=
−
e
−
v
x
2
π
|
x
2
2
∞
=
ϕ
(
x
)
x
.
{\displaystyle Q(x)=\int _{x}^{\infty }\phi (u)\,du<\int _{x}^{\infty }{\frac {u}{x}}\phi (u)\,du=\int _{\frac {x^{2}}{2}}^{\infty }{\frac {e^{-v}}{x{\sqrt {2\pi }}}}\,dv=-{\biggl .}{\frac {e^{-v}}{x{\sqrt {2\pi }}}}{\biggr |}_{\frac {x^{2}}{2}}^{\infty }={\frac {\phi (x)}{x}}.}
Similarly, using
ϕ
′
(
u
)
=
−
u
ϕ
(
u
)
{\displaystyle \phi '(u)=-u\phi (u)}
and the quotient rule ,
(
1
+
1
x
2
)
Q
(
x
)
=
∫
x
∞
(
1
+
1
x
2
)
ϕ
(
u
)
d
u
>
∫
x
∞
(
1
+
1
u
2
)
ϕ
(
u
)
d
u
=
−
ϕ
(
u
)
u
|
x
∞
=
ϕ
(
x
)
x
.
{\displaystyle \left(1+{\frac {1}{x^{2}}}\right)Q(x)=\int _{x}^{\infty }\left(1+{\frac {1}{x^{2}}}\right)\phi (u)\,du>\int _{x}^{\infty }\left(1+{\frac {1}{u^{2}}}\right)\phi (u)\,du=-{\biggl .}{\frac {\phi (u)}{u}}{\biggr |}_{x}^{\infty }={\frac {\phi (x)}{x}}.}
Solving for Q (x ) provides the lower bound.
Q
(
x
)
≤
e
−
x
2
2
,
x
>
0
{\displaystyle Q(x)\leq e^{-{\frac {x^{2}}{2}}},\qquad x>0}
Improved exponential bounds and a pure exponential approximation are [5]
Q
(
x
)
≤
1
4
e
−
x
2
+
1
4
e
−
x
2
2
≤
1
2
e
−
x
2
2
,
x
>
0
{\displaystyle Q(x)\leq {\tfrac {1}{4}}e^{-x^{2}}+{\tfrac {1}{4}}e^{-{\frac {x^{2}}{2}}}\leq {\tfrac {1}{2}}e^{-{\frac {x^{2}}{2}}},\qquad x>0}
Q
(
x
)
≈
1
12
e
−
x
2
2
+
1
4
e
−
2
3
x
2
,
x
>
0
{\displaystyle Q(x)\approx {\frac {1}{12}}e^{-{\frac {x^{2}}{2}}}+{\frac {1}{4}}e^{-{\frac {2}{3}}x^{2}},\qquad x>0}
A tight approximation for whole range of positive arguments is given by Karagiannidis & Lioumpas (2007) [6] [failed verification ]
Q
(
x
)
≈
(
1
−
e
−
1.4
x
)
e
−
x
2
2
1.135
2
π
x
,
x
>
0
{\displaystyle Q(x)\approx {\frac {\left(1-e^{-1.4x}\right)e^{-{\frac {x^{2}}{2}}}}{1.135{\sqrt {2\pi }}x}},x>0}
Inverse Q
The inverse Q -function can be trivially related to the inverse error function:
Q
−
1
(
x
)
=
2
e
r
f
−
1
(
1
−
2
x
)
{\displaystyle Q^{-1}(x)={\sqrt {2}}\ \mathrm {erf} ^{-1}(1-2x)}
Values
The Q -function is well tabulated and can be computed directly in most of the mathematical software packages such as Matlab and Mathematica. Some values of the Q -function are given below for reference.
Q (0.0) = 0.500000000 = 1/2.0000
Q (0.1) = 0.460172163 = 1/2.1731
Q (0.2) = 0.420740291 = 1/2.3768
Q (0.3) = 0.382088578 = 1/2.6172
Q (0.4) = 0.344578258 = 1/2.9021
Q (0.5) = 0.308537539 = 1/3.2411
Q (0.6) = 0.274253118 = 1/3.6463
Q (0.7) = 0.241963652 = 1/4.1329
Q (0.8) = 0.211855399 = 1/4.7202
Q (0.9) = 0.184060125 = 1/5.4330
Q (1.0) = 0.158655254 = 1/6.3030
Q (1.1) = 0.135666061 = 1/7.3710
Q (1.2) = 0.115069670 = 1/8.6904
Q (1.3) = 0.096800485 = 1/10.3305
Q (1.4) = 0.080756659 = 1/12.3829
Q (1.5) = 0.066807201 = 1/14.9684
Q (1.6) = 0.054799292 = 1/18.2484
Q (1.7) = 0.044565463 = 1/22.4389
Q (1.8) = 0.035930319 = 1/27.8316
Q (1.9) = 0.028716560 = 1/34.8231
Q (2.0) = 0.022750132 = 1/43.9558
Q (2.1) = 0.017864421 = 1/55.9772
Q (2.2) = 0.013903448 = 1/71.9246
Q (2.3) = 0.010724110 = 1/93.2478
Q (2.4) = 0.008197536 = 1/121.9879
Q (2.5) = 0.006209665 = 1/161.0393
Q (2.6) = 0.004661188 = 1/214.5376
Q (2.7) = 0.003466974 = 1/288.4360
Q (2.8) = 0.002555130 = 1/391.3695
Q (2.9) = 0.001865813 = 1/535.9593
Q (3.0) = 0.001349898 = 1/740.7967
Q (3.1) = 0.000967603 = 1/1033.4815
Q (3.2) = 0.000687138 = 1/1455.3119
Q (3.3) = 0.000483424 = 1/2068.5769
Q (3.4) = 0.000336929 = 1/2967.9820
Q (3.5) = 0.000232629 = 1/4298.6887
Q (3.6) = 0.000159109 = 1/6285.0158
Q (3.7) = 0.000107800 = 1/9276.4608
Q (3.8) = 0.000072348 = 1/13822.0738
Q (3.9) = 0.000048096 = 1/20791.6011
Q (4.0) = 0.000031671 = 1/31574.3855
References
^ The Q-function , from cnx.org
^ a b Basic properties of the Q-function
^ Normal Distribution Function - from Wolfram MathWorld
^ John W. Craig, A new, simple and exact result for calculating the probability of error for two-dimensional signal constellaions , Proc. 1991 IEEE Military Commun. Conf., vol. 2, pp. 571–575.
^ Chiani, M., Dardari, D., Simon, M.K. (2003). New Exponential Bounds and Approximations for the Computation of Error Probability in Fading Channels . IEEE Transactions on Wireless Communications, 4(2), 840–845, doi=10.1109/TWC.2003.814350.
^ Karagiannidis, G. K., & Lioumpas, A. S. (2007). An improved approximation for the Gaussian Q-function . Communications Letters, IEEE, 11(8), 644-646.