Fundamentals of Probability and Statistics for Engineers

Considerable importance is attached to the results expressed by Equations
(5.54) and (5.55) because sums of random variables occur frequently in prac-
tical situations. By way of recognizing this fact, Equation (5.55) is repeated now
as Theorem 5.3.

Theorem 5. 3. Let Y X 1 X 2 ,andletX 1 and X 2 be independent and con-
tinuous random variables. Then the pdf of Y is the convolution of the pdfs
associated with X 1 and X 2 ;thatis,

Repeated applications of this formula determine fY (y) when Y is a sum of
any number of independent random variables.

Ex ample 5. 16. Problem: determine fY (y) of Y X 1 X 2 when X 1 and X 2 are
independent and identically distributed according to

and similarly for X 2.
Answer: Equation (5.56) in this case leads to

x 2

x 1

x 1 +x 2 =y

y

y

R^2

Figure 5. 20 Region R^2 :x 1 x 2 y

146 Fundamentals of Probability and Statistics for Engineers

ˆ ‡

fY…y†ˆ

Z 1

1

fX 1 …y x 2 †fX 2 …x 2 †dx 2 ˆ

Z 1

1

fX 2 …y x 1 †fX 1 …x 1 †dx 1 : … 5 : 56 †

ˆ‡

fX 1 …x 1 †ˆ

ae^ ax^1 ; forx 1  0 ;

0 ; elsewhere;



… 5 : 57 †

fY…y†ˆa^2

Z y

0

e^ a…y^ x^2 †e^ ax^2 dx 2 ; y 0 ; … 5 : 58 †

‡