Random Variable - Random Signals and Processes Primer with MATLAB - page 47

Using ( 2.71 ), we arrive at:

f X ðxÞ¼ 1
i¼1

Pfx i Þdðx x i Þ: (2.74)

Example 2.3.2 In this example, we are looking for the density function of
the discrete variable from Example 2.2.2, where the distribution was F X ( x ) ¼
1/6 u ( x )+ ... + 1/6 u ( x 6). Find the probability that the random variable is also
less than 4.

Solution Using ( 2.74 ), we have the density

6 X

6

f X ðxÞ¼ 1 = dðx kÞ: (2.75)

k¼ 1

This density is shown in Fig. 2.16 . The desired probability (not including
point 4) is:

ð

3

ð

3

X

6 X

6 6

1
6

1 1

6 ð 1 þ 1 þ 1 Þ¼

1
2 :

PfX < 4 g¼ dðx kÞ d x ¼ dðx kÞ d x ¼

k¼ 1 k¼ 1

1 1

(2.76)

If we include point 4, we have:

ð

4

ð

4

X

6 X

6 6

1
6

1

PfX 4 g¼ dðx kÞ d x ¼ dðx kÞ d x

(2.77)

k¼ 1 k¼ 1

1 1

1
6 ð 1 þ 1 þ 1 þ 1 Þ¼

2
3 :

¼

Fig. 2.16 Illustration of
PDF for a discrete r.v.