Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 214

(c) Find the mathematical PDF of Y as the convolution of the corresponding PDFs,

f Y ðyÞ ¼ f X 1 ðx 1 Þf X 2 ðx 2 Þ: (3.334)

Solution The plot in Fig. 3.33a indicates that the variables are independent. Their
sum is shown in Fig. 3.32b .
The estimated and the mathematical PDFs are shown in Fig. 3.34 .

a b

X1+X2

2 8

7

1.5

6

1

5

0.5

4

0

3

-0.5

2

-1

1

-1.5

0

-2 -1

1 2 3 4 5 6 0 200 400 600 800 1000

X1

Fig. 3.33 Independent variables and their sum in Exercise M.3.4

X 1+X2

0.2

0.15

0.1

0.05

0

-1 0 1 2 3 4 5 6 7 8

cells

CONVOLUTION OF PD F s

0.2

0.15

0.1

0.05

0

-1 0 1 2 3 4 5 6 7 8

range

Fig. 3.34 Estimated and mathematical PDFs in Exercise M.3.4