From Wikipedia, the free encyclopedia
A plot of the Q-function. In statistics , the Q-function is the tail probability of the standard normal distribution .[1] [2]
Q
(
x
)
{\displaystyle Q(x)}
x
{\displaystyle x}
cumulative distribution function , are also used occasionally.[3]
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 {\Bigl (}-{\frac {u^{2}}{2}}{\Bigr )}\,du.}
Thus,
Q
(
x
)
=
Q
(
−
x
)
=
1
−
Φ
(
x
)
,
{\displaystyle Q(x)=Q(-x)=1-\Phi (x)\,\!,}
where
Φ
(
x
)
{\displaystyle \Phi (x)}
cumulative distribution function of the normal Gaussian distribution .
The Q-function can be expressed in terms of the error function as[2]
Q
(
x
)
=
1
2
−
1
2
erf
(
x
2
)
=
1
2
erfc
(
x
2
)
.
{\displaystyle Q(x)={\tfrac {1}{2}}-{\tfrac {1}{2}}\operatorname {erf} {\Bigl (}{\frac {x}{\sqrt {2}}}{\Bigr )}={\tfrac {1}{2}}\operatorname {erfc} ({\frac {x}{\sqrt {2}}}).}
Bounds
x
1
+
x
2
⋅
1
2
π
e
−
x
2
/
2
<
Q
(
x
)
<
1
x
⋅
1
2
π
e
−
x
2
/
2
,
x
>
0
,
{\displaystyle {\frac {x}{1+x^{2}}}\cdot {\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}<Q(x)<{\frac {1}{x}}\cdot {\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2},\qquad x>0,}
become increasingly tight for large x , and are often useful.
Using the substitution
v
=
u
2
/
2
{\displaystyle v=u^{2}/2}
φ
(
x
)
=
1
2
π
e
−
x
2
/
2
,
{\displaystyle \varphi (x)={\tfrac {1}{\sqrt {2\pi }}}e^{-x^{2}/2},}
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 {\begin{aligned}Q(x)&=\int _{x}^{\infty }\varphi (u)\,du\\&<\int _{x}^{\infty }{\frac {u}{x}}\varphi (u)\,du=\int _{x^{2}/2}^{\infty }{\frac {e^{-v}}{x{\sqrt {2\pi }}}}\,dv=-{\biggl .}{\frac {e^{-v}}{x{\sqrt {2\pi }}}}{\biggr |}_{x^{2}/2}^{\infty }={\frac {\varphi (x)}{x}}.\end{aligned}}}
Similarly, using
φ
′
(
u
)
=
−
u
φ
(
u
)
{\displaystyle \scriptstyle \varphi '(u)\,{=}\,-u\,\varphi (u)}
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 {\begin{aligned}{\Bigl (}1+{\frac {1}{x^{2}}}{\Bigr )}Q(x)&=\int _{x}^{\infty }{\Bigl (}1+{\frac {1}{x^{2}}}{\Bigr )}\varphi (u)\,du\\&>\int _{x}^{\infty }{\Bigl (}1+{\frac {1}{u^{2}}}{\Bigr )}\varphi (u)\,du=-{\biggl .}{\frac {\varphi (u)}{u}}{\biggr |}_{x}^{\infty }={\frac {\varphi (x)}{x}}.\end{aligned}}}
Solving for
Q
(
x
)
{\displaystyle Q(x)}
Q
(
x
)
≤
1
2
e
−
x
2
2
,
x
>
0
{\displaystyle {\begin{aligned}Q(x)\leq {\frac {1}{2}}e^{-{\frac {x^{2}}{2}}},\qquad x>0\end{aligned}}}
Values
The Q -function is well tabulated and can be computed directly in most of the mathematical software packages such as Matlab, Mathematica. Some values of the Q -function are given below for reference.
Q (0.0) = 0.500000000Q (0.1) = 0.460172163Q (0.2) = 0.420740291Q (0.3) = 0.382088578Q (0.4) = 0.344578258Q (0.5) = 0.308537539Q (0.6) = 0.274253118Q (0.7) = 0.241963652Q (0.8) = 0.211855399Q (0.9) = 0.184060125
Q (1.0) = 0.158655254Q (1.1) = 0.135666061Q (1.2) = 0.115069670Q (1.3) = 0.096800485Q (1.4) = 0.080756659Q (1.5) = 0.066807201Q (1.6) = 0.054799292Q (1.7) = 0.044565463Q (1.8) = 0.035930319Q (1.9) = 0.028716560
Q (2.0) = 0.022750132Q (2.1) = 0.017864421Q (2.2) = 0.013903448Q (2.3) = 0.010724110Q (2.4) = 0.008197536Q (2.5) = 0.006209665Q (2.6) = 0.004661188Q (2.7) = 0.003466974Q (2.8) = 0.002555130Q (2.9) = 0.001865813
Q (3.0) = 0.001349898Q (3.1) = 0.000967603Q (3.2) = 0.000687138Q (3.3) = 0.000483424Q (3.4) = 0.000336929Q (3.5) = 0.000232629Q (3.6) = 0.000159109Q (3.7) = 0.000107800Q (3.8) = 0.000072348Q (3.9) = 0.000048096Q (4.0) = 0.000031671
See also
References
