The QQ-function is a convenient way to
express right-tail probabilities for normal (Gaussian) random
variables. For
x∈R
x
,
Qx
Q
x
is defined as the probability that a standard normal
random variable (zero mean, unit variance) exceeds
xx:
Qx=12π∫x∞e−t22d
t
Q
x
1
2
t
x
t
2
2
QQ is a mapping from
R to
01
0
1
. One may also define
Q−∞=1
Q
1
and
Q∞=0
Q
0
.
Some authors define the
QQ-function in different
ways. One alternative is to define
Qx=Fx−F0
Q
x
F
x
F
0
. This definition is discussed at
MathWorld.
Since
Qx
Q
x
is monotonically decreasing, it has a well-defined inverse
Q-1
:
01
→
R
Q
:
0
1
→
.
If
Fx
F
x
denotes the cumulative distribution function of a
standard normal, then clearly
Qx=1−Fx
Q
x
1
F
x
. For this reason
QQ is also called the
complementary cumulative distribution function. The
QQ-function is useful
because the tail probability cannot be evaluated symbolically,
and so
Qx
Q
x
offers a concise notation for this integral. It is
similar to the gamma and beta functions in this respect.
The QQ-function is also useful for
expressing right-tail probabilities of nonstandard normal
variates. If
X∼𝒩μσ2
X
μ
σ
2
then
X−μσ∼𝒩01
X
μ
σ
0
1
To express
PrX>γ
X
γ
in terms of QQ, where
γ∈R
γ
, define
η=γ−μσ
η
γ
μ
σ
. Then
PrX>γ=PrX>ησ+μ=PrX−μσ>η=Qη=Qγ−μσ
X
γ
X
η
σ
μ
X
μ
σ
η
Q
η
Q
γ
μ
σ
(1)The erferf
function is defined as
erfx=2π∫0xe−t2d
t
erf
x
2
t
0
x
t
2
Both erferf and its
inverse, erfinverfinv, are
built into many common mathematical software packages such as
Mathematica and Matlab. Therefore, they can be used to
numerically evaluate QQ and
Q-1
Q
. By a change of variables, we have
Qx=12(1−erfx2)
Q
x
1
2
1
erf
x
2
and
Q-1y=2erfinv1−2y
Q
y
2
erfinv
1
2
y
One approximation that is sometimes useful for
xx away from zero is
Qx≃1x2πe-12x2
Q
x
1
x
2
-1
2
x
2
