Last version available at www.eng.tau.ac.il/∼jo/teaching
Q function and error function
We first note that
Z
∞
−∞
e
−x
2
dx =
√
π ;
Z
∞
−∞
e
−
ax
2
2
dx =
r
2π
a
For our needs in Digital Communication course, we define:
Q(α)
∆
=
1
√
2π
Z
∞
α
e
−
x
2
2
dx
The Q(·) function is monotonically decreasing. Some features:
Q(−∞) = 1 ; Q(0) =
1
2
; Q(∞) = 0 ; Q(−x) = 1 − Q(x)
Known bounds (valid for x > 0):
1
√
2πx
µ
1 −
1
x
2
¶
e
−x
2
/2
< Q(x) <
1
√
2πx
e
−x
2
/2
Q(x) ≤
1
2
e
−x
2
/2
Matlab does not have a build-in function for Q(·). Instead, we use its erf function:
erf(α)
∆
=
2
√
π
Z
α
0
e
−x
2
dx
Note that erf function is defined over [0, ∞) only, and
erf(0) = 0 ; erf(∞) = 1
The relations between the two functions are
Q(α) =
1
2
−
1
2
erf
µ
α
√
2
¶
; erf(α) = 1 − 2Q(
√
2α)
If we have a normal variable X ∼ N (µ, σ
2
), the probability that X > x is
Pr{X > x} = Q
µ
x − µ
σ
¶
Now, if we want to know the probability of X to be away from its expectation µ by at least a
(either to the left or to the right) we have:
Pr{X > µ + a} = Pr{X < µ − a} = Q
µ
a
σ
¶
The probability to be away from the center where we don’t matter in which direction is 2 ·Q(
a
σ
).
This version compiled on April 6, 2006