1
Q
function Handout –
c
±
Keith M. Chugg
1
Deﬁnitions
The
Q
function is tail integral of a unitGaussian pdf, and is deﬁned as
Q
(
z
)
Δ
=
Z
∞
z
1
√
2
π
e

x
2
2
dx.
The
Q
function has the following properties:
lim
z
→∞
Q
(
z
)=0
lim
z
→∞
Q
(
z
)=1
Q
(0)=1
/
2
Q
(

z

Q
(
z
)
.
There are several other common notations used to denote this integral function or a close
relative. The
Q
function is sometime referred to as the “Gaussian Integral Function” and
denoted GIF(
z
). Other functions which are closely related are the erf(
·
) (error function) and
erfc(
·
) (complementary error function):
erf(
z
)
Δ
=
Z
z
0
2
√
π
e

x
2
dx z
≥
0
erfc(
z
)
Δ
=
Z
∞
z
2
√
π
e

x
2
dx
=1

erf(
z
)
z
≥
0
The
Q
function is related to these functions by
Q
(
z
)=
1
2
"
1

erf
ˆ
z
√
2
!#
=
1
2
erfc
ˆ
z
√
2
!
z
≥
0
.
It is clear that if
X
(
u
) is a mean zero, unit variance Gaussian random variable, that
Q
(
z

F
X
(
u
)
(
z
)
.
A useful relation is that if
Y
(
u
) is Gaussian with mean
m
and variance
σ
2
, then
Pr
{
Y
(
u
)
>a
}
=
Q
±
a

m
σ
¶
.
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 Spring '07
 ToddBrun
 Numerical Analysis, Normal Distribution, Error function, e−x dx, Qfunction, Gaussian Integral Function

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