Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 235

4.2.2 Practical Calculation

4.2.2.1 Error Function

The error function is denoted as erf( x ) and is defined in the literature in different
ways. We adopt the definition from [SCH75, p. 89]:

ð

x

2
p e u 2 d u: erf ðxÞ¼ p (4.23)

0

This function has some important properties which can be easily verified from
the integral ( 4.23 ):

erf ð 0 Þ¼ 0 ; (4.24a)

erf ð1Þ ¼ 1 ; (4.24b)

erf ð1Þ ¼ 1 ; (4.24c)

erf ðxÞ¼ erf ðxÞ: (4.24d)

In the following, we show how the normal distribution can be expressed in terms
of the erf function. To this end, in the integral ( 4.22 ) we introduce the variable

x m
p s

u ¼ (4.25)

resulting in:

2 3

ð ð ð

u 0 u

1
p

1
2

2
p

2
p e u 2 d u ¼

4

e u 2 d u þ e u 2 d u

5

F X ðxÞ¼ p p p

1 1

0

2 3

ð 1 ð

u

1
2

2
p

2
p

4

e u 2 d u þ e u 2 d u

5 :

¼ p p (4.26)

0 0

Applying the definition ( 4.23 ) and property ( 4.24c ), the first term in ( 4.26 )
becomes:

ð 1

2
p

e u 2 d u ¼ erf ð1Þ ¼ ð 1 Þ¼ 1 p : (4.27)

0