Random Variable - Random Signals and Processes Primer with MATLAB - page 92

placing ( 2.289 ) into ( 2.261 ), we have:

1

x X
i

m ¼ PfX ¼ x i gdðx x i Þ d x: (2.290)

1

Interchanging the sum and the integral in ( 2.290 ), and knowing that

1

x i for x ¼ x i ;

xdðx x i Þ d x ¼ (2.291)

0 otherwise :

1

we arrive at:

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m ¼ X
i

xdðx x i Þ d x ¼ X
i

PfX ¼ x i g x i PfX ¼ x i g: (2.292)

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This is the same result as in ( 2.223 ).
To this end, we will use the expression ( 2.261 ) for the mean value in order to
explain different characteristics of randomvariables considered in the rest of the topic.

Example 2.7.13 We want to find the mean value of the discrete variable X from
Example 2.7.2 using ( 2.261 ).

Solution The density function is:

f X ðxÞ¼ 0 : 5 dðxÞþ 0 : 5 dðx 1 Þ: (2.293)

From ( 2.261 ), we have:

1 1

m X ¼ xf X ðxÞ d x ¼ x 0 ½ :

5 dðxÞþ 0

:

5 dðx 1 Þ

d x;

1 1

(2.294)

1 1

¼ 0 : 5 xdðxÞ d x þ 0 : 5 xdðx 1 Þ d x:

1 1

The first integral on the right side of ( 2.294 ) is:

1

xdðxÞ d x ¼ 0 (2.295)

1