Random Variable - Random Signals and Processes Primer with MATLAB - page 89

Fig. 2.44 Density function in Example 2.7.11

Solution From ( 2.273 ), we have:

ð ð

2 2

1
3 d x ¼ 3

EfgðXÞg ¼ Ef 2 X 2 ð 2 x 2

þ 1 g¼ gðxÞf X ðxÞ d x ¼ þ 1 Þ : (2.275)

1 1

2.7.3.1 Properties

We can easily verify that the properties considered for the discrete random variable
also stand for a continuous variable.

P.1 The mean value of the constant is the constant itself.

Consider the expected value of the constant b . In this case, there is no random
variable but only one discrete value b with the probability P { b } ¼ 1.
The corresponding density function is a delta function dðx bÞ (Fig. 2.45 )

f X ðxÞ¼dðx bÞ: (2.276)

From ( 2.261 ), we have:

1 1

Efbg¼ xf X ðxÞ d x ¼ xdðx bÞ d x: (2.277a)

1 1

Fig. 2.45 Density function of the constant b