Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 209

Fig. 3.26 Convolution of the uniform PDFs

Exercise 3.18 The random variables X 1 and X 2 are independent. Find the density of
their sum if the corresponding densities are given as:

f X 1 ðx 1 Þ¼a e ax 1 uðx 1 Þ; f X 2 ðx 2 Þ¼b e bx 2 uðx 2 Þ; (3.322)

where a and b are constants.

Solution The density of the sum

X ¼ X 1 þ X 2 (3.323)

is equal to the convolution of the densities ( 3.322 ):

f X ðxÞ¼f X 1 ðx 1 Þf X 2 ðx 2 Þ: (3.324)

For x < 0,

f X ðxÞ¼ 0 : (3.325)

For x > 0,

1

f X ðxÞ¼ f X 1 ðx 1 Þf X 2 ðx x 1 Þ d x 1

1

¼ ab 1

ab
a b e ax

e ax 1 e bðxx 1 Þ d x 1 ¼

: (3.326)

x

Exercise 3.19 Find the characteristic function of the variable X with the density
function shown in Fig. 3.27 in terms of the characteristic function of the variables
X 1 and X 2 . The variable X is the sum of X 1 and X 2 , and the variables X 1 and X 2 are
independent.