Random Variable - Random Signals and Processes Primer with MATLAB - page 51

Taking a ¼ 1, and using ( 2.89 ), the density ( 2.87 ) becomes:

f X ðxÞ¼ e 2 jj

; 1< x <1:

(2.90)

The distribution function is obtained from the PDF using ( 2.61 ).
For x 0

ð ð

x x

e 2 x
2 :

e 2 x d x ¼

F X ðxÞ¼ f X ðxÞ d x ¼ (2.91)

1 1

For x 0

ð ð ð

x 0 x

1 e 2 x
2

e 2 x
2 :

1
2 þ

e 2 x d x ¼

e 2 x d x þ

F X ðxÞ¼ f X ðxÞ d x ¼ ¼ 1 (2.92)

1 1

0

The PDF and distribution are given in Fig. 2.19 .
The probability that the X is negative is related to the PDF as:

ð

0

1
2 :

e 2 x d x ¼

PfX 0 g¼ (2.93)

1

Similarly, the probability and the distribution are related as:

PfX 0 g¼F X ð 0 Þ: (2.94)

Taking either ( 2.91 )or( 2.92 ), we have:

e 0
2 ¼

1
2 :

PfX 0 g¼F X ð 0 Þ¼ 1 (2.95)

Fig. 2.19 Illustration of PDF, distribution, and probabilities from Example 2.3.4. ( a ) PDF.
( b ) Distribution