Last version available at www.eng.tau.ac.il/∼jo/teaching
Q function and error function
We first note that
Z∞
−∞
e−x2dx =√π;Z∞
−∞
e−ax2
2dx =r2π
a
For our needs in Digital Communication course, we define:
Q(α)∆
=1
√2πZ∞
α
e−x2
2dx
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 µ1−1
x2¶e−x2/2< Q(x)<1
√2πx e−x2/2
Q(x)≤1
2e−x2/2
Matlab does not have a build-in function for Q(·). Instead, we use its erf function:
erf(α)∆
=2
√πZα
0
e−x2dx
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
2erf µα
√2¶; erf(α) = 1 −2Q(√2α)
If we have a normal variable X∼N(µ, σ2), the probability that X > x is
Pr{X > x}=Qµx−µ
σ¶
Now, if we want to know the probability of Xto be away from its expectation µby at least a
(either to the left or to the right) we have:
Pr{X > µ +a}= Pr{X < µ −a}=Qµa
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