Random Variable - Random Signals and Processes Primer with MATLAB - page 44

we have:

Pfx <Xx þ Dxg
Dx

Pfx <Xx þ d xg
d x

f X ðxÞ ¼ lim
Dx! 0

¼ : (2.64)

This expression represents the probability that a random variable X lies in an
infinitesimal interval around the point x , normalized by the length of the interval.
Hence, the name density comes.
Avisualization of this expression is given in Fig. 2.13 , assuming f X ( x ) ¼ f X ( x + Dx ).
Example 2.3.1 Find the density function of the continuous variable from Example
2.2.3, assuming that a ¼ 2 and b ¼ 6. Also find the probability that the random
variable is less than 4.

Solution For convenience, the distribution ( 2.30 ) is presented again in Fig. 2.14a
for the given values a and b .
Using ( 2.60 ), we have:

8
<

0 for x < 2 ;

¼

x 2
4

1
4

d x

f X ðxÞ ¼ for 2 x 6 ; (2.65)

:

0 for x > 6 :

Fig. 2.13 Visualization of density functions

Fig. 2.14 Illustration of Example 2.3.1. ( a ) Distribution. ( b ) PDF