Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 245

a b

LOGARITHMIC TRANSFORMATION LOGARITHMIC TRANSFORMATION OF NORMAL R.V.

0.8

60

Y=e X
INPUT NORMAL VARIABLE:
m=1, VA=.25

0.7

50

0.6

40

0.5

0.4

30

0.3

20

0.2

OUTPUT VARIABLE

10

0.1

0

0

−2 0 2 4 6 8 10 12 14 16

-2 -1 0 1 2 3 4

X Range

Fig. 4.8 Logarithmic transformation. ( a ) Transformation. ( b ) PDFs

Solution The transformation is given as:

X for X> 0 ;

Y ¼ jj¼ (4.79)

X for X< 0 :

From here

¼ 1

d y
d x

: (4.80)

Using ( 2.170 ) and ( 4.80 ), we have:

2
2 p

e y 2 f Y ðyÞ¼f X ðxÞþf X ðxÞj y¼ jj ¼ 2 f X ðyÞ¼ p ; y> 0 : (4.81)

The transformation as well as the input and output PDFs are shown in Fig. 4.9a ,b,
respectively.
Next example illustrates how the discrete random variable is obtained from the
transformation of a normal random variable.
Example 4.3.4 The transformation of a normal random variable X with a mean
value equal to 0 and a variance equal to 1, is given as:

0 : 3 for X> 0 ;

Y ¼ (4.82)

0 : 3 for X< 0 :

Find the PDF of the random variable Y .