Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 238

Similarly, the probability ( 4.31 ) can be expressed in terms of complementary
error function as:

Pfx 1 X x 2 g¼F X ðx 2 ÞF X ðx 1 Þ

1 þ erfc

1
2

x 2 m
p s

x 1 m
p s

¼ 1 erfc

erfc

1
2

x 1 m
p s

x 2 m
p s

¼ erfc : (4.41)

The probability ( 4.33 ) is:

¼ 1 erfc :

k k

Pfm ks X m þ ksg¼ erf p 2 p 2 (4.42)

Example 4.2.2 Find the probabilities from Example 4.2.1 in terms of the erfc
function, where the erfc function is calculated using the MATLAB file erfc( x ) .
Solution From ( 4.40 ) and Example 4.2.1, we have:

1
2 erfc

x 1 m
p s PfX 10 g¼ 1

¼ 1

1
2 erfc

5
8 2

1
2 0

¼ 1 p : 532 ¼ 0 : 734 : (4.43)

From ( 4.41 ), we get:

Pf 2 X 8 g

erfc ¼

1
2

erfc 7

3
8 2

1
2 1

¼

8 2

p p ½ : 6184 0 : 7077 ¼ 0 : 4554 : (4.44)

4.2.2.3 Q Function

Consider the probability that the values of a normal random variable are outside of
the interval m + ks ,

1 1

1
2 p

e ð x m Þ 2 PfX>m þ ksg¼ f X ðxÞ d x ¼ p d x: (4.45)

2 s 2

s

mþks mþks

Introducing the variable

x m
s

u ¼ (4.46)