The Complementary Unit Gaussian Distribution Function
Q
(
x
)
Let
φ
(
u
), where
φ
(
u
) =
1
√
2
π
exp

u
2
2
,
denote the probability density function (pdf)
of a standard (or unit) Gaussian random variable. The cumulative probability distribution
function (CDF) of this random variable is denoted Φ(
x
) where
Φ(
x
) =
Z
x
∞
φ
(
u
)
du
=
Z
x
∞
1
√
2
π
exp

u
2
2
du.
Φ(
x
) is also known as the
unit Gaussian distribution function
and
Q
(
x
) = 1

Φ(
x
) =
Z
∞
x
φ
(
u
)
du
=
Z
∞
x
1
√
2
π
exp

u
2
2
du
is called the
complementary unit Gaussian distribution function.
Note that since Φ(
x
) is a
monotone
increasing
function rising from 0 at
∞
to 1 at
∞
,
Q
(
x
) is a monotone
decreasing
function falling from 1 at
∞
to 0 at
∞
.
In many applications in communications and
signal processing,
Q
(
x
) is slightly more convenient to use than Φ(
x
). For example, the bit
error rate in some communication systems can be expressed as
Q
(
√
SNR) where SNR is the
signaltonoise ratio
. Since
Q
(
x
) is a decreasing function of its argument, maximizing the
SNR is an important objective in communication system design.
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 Spring '09
 Normal Distribution, Probability distribution, Probability theory, probability density function, Cumulative distribution function

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