Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 240

Example 4.2.3 Find the probabilities of Example 4.2.1 in terms of Q function.

Solution From ( 4.53 ), it follows:

:

5
8

PfX 10 g¼ 1 PfX> 10 g¼ 1 QðkÞ¼ 1 Q (4.56)

Using ( 4.54 ), we arrive at:

Q :

7
8

3
8

Pf 2 <X 8 g¼ 1 Q (4.57)

4.2.2.4 Relation Between Different Functions

Sometimes it is necessary to express the probability (given in terms of one function)
in terms of a different function.
From ( 4.42 ) and ( 4.55 ), we write:

¼ 1 2 QðxÞ:

x

p 2

erf (4.58)

From here, we have:

1
2

x
2

QðxÞ¼ 1 erf p : (4.59)

p x

erf ðxÞ¼ 1 2 Q : (4.60)

Similarly, from ( 4.42 ) and ( 4.55 ), we have:

¼ 1 2 QðxÞ;

x

p

1 erfc 2 (4.61)

resulting in:

1
2 erfc

x

QðxÞ¼ p 2 (4.62)

and

p x

erfc ðxÞ¼ 2 Q : (4.63)

The relations ( 4.38 ), ( 4.59 ), ( 4.60 ), ( 4.62 ), and ( 4.63 ) are summarized in
Table 4.1 .