Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 192

Using ( 2.387 ), we get:

1

1
2 p

f X 2 ðoÞ e joðyx 1 Þ d o ¼ f X 2 ðy x 1 Þ;

(3.214)

1

From ( 3.213 ) and ( 3.214 ), we arrive at:

1

f Y ðyÞ¼ f X 1 ðx 1 Þf X 2 ðy x 1 Þ d x 1 : (3.215)

1

This expression can be rewritten as:

1

f Y ðyÞ¼ f X 2 ðx 2 Þ f X 1 ðy x 2 Þ d x 2 : (3.216)

1

The expressions ( 3.215 ) and ( 3.216 ) present the convolution of the PDFs of
random variables X 1 and X 2 , and can be presented as:

f Y ðyÞ¼f X 1 ðx 1 Þf X 2 ðx 2 Þ¼f X 2 ðx 2 Þf X 1 ðx 1 Þ; (3.217)

where * stands for the convolution operation.
This result can be generalized for the sum of N independent variables X i ,

Y ¼ X

N

X i ; (3.218)

i¼ 1

f Y ðyÞ¼f X 1 ðx 1 Þf X 2 ðx 2 Þf X N ðx N Þ: (3.219)

Example 3.5.3 Two resistors R 1 and R 2 are in a serial connection (Fig. 3.16a ). Each
of them randomly changes its value in a uniform way for 10% about its nominal
value of 1,000

. Find the PDF of the equivalent resistor R ,

O

R ¼ R 1 þ R 2 (3.220)

if both resistors have uniform density in the interval [900, 1,100] .

O

Solution Denote the equivalent resistor R as a random variable X and the particular
resistors R 1 and R 2 , as the random variables X 1 and X 2 . Then,

X ¼ X 1 þ X 2 : (3.221)