Random Variable - Random Signals and Processes Primer with MATLAB - page 50

Fig. 2.18 PDF and probability

Example 2.3.4 The PDF of a variable X is given as

f X ðxÞ¼a e bjj

; 1< x <1; (2.87)

where both a and b are constant. Find the relation between a and b . Find the
corresponding distribution as well as the probability that the variable is negative,
if a ¼ 1. Relate the probability to both the PDF and distribution. Do the same for
the probability that the variable is in the interval [1, 2].

Solution The PDF must satisfy the property ( 2.83 ):

1 1 ð

0

1

a
b þ

a
b ¼

2 a
b :

a e bjj d x ¼ a e bx d xþ a e bx d x ¼

f X ðxÞ d x ¼ 1 ¼ (2.88)

1 1 1 0

From ( 2.88 ), we have:

b ¼ 2 a: (2.89)