Fundamentals of Probability and Statistics for Engineers

determination of the distribution Y when all Xj, j 1,2,...,n, are continuous
random variables. Consider the transformation

where the joint distribution of X 1 ,X 2 ,...,andXn is assumed to be specified in
term of their joint probability density function (jpdf), fX 1 ...Xn (x 1 ,...,xn), or
their joint probability distribution function (JPDF), FX 1 ...Xn (x 1 ,...,xn). In a
more compact notation, they can be written as fX ( x)andFX ( x), respectively,
where X is an n-dimensional random vector with co mponents X 1 ,X 2 ,...,Xn.
The starting point of the derivation is the same as in the single-random-
variable case; that is, we consider FY (y) P(Y y). In terms of X,this
probability is equal to P[g( X) y]. Thus:

The final expression in the above represents the JPDF ofXfor which the

X

where the limits of the integrals are determined by an n-dimensional region Rn
within which g( x) y is satisfied. In view of Equations (5.41) and (5.42), the
PDF of Y, FY (y), can be determined by evaluating the n-dimensional integral in
Equation (5.42). The crucial step in this derivation is clearly the identification
of Rn, which must be carried out on a problem-to-problem basis. As n becomes
large, this can present a formidable obstacle.
The procedure outlined above can be best demonstrated through examples.
Ex ample 5. 11. Problem: let Y X 1 X 2. Determine the pdf of Y in terms of
fX 1 X 2 (x 1 ,x 2 ).
Answer: from Equations (5.41) and (5.42), we have

The equation x 1 x 2 y is graphed in Figure 5.16 in which the shaded area
represents R^2 ,orx 1 x 2 y. The limits of the double integral can thus be
determined and Equation (5.43) becomes

138 Fundamentals of Probability and Statistics for Engineers

ˆ

Yˆg…X 1 ;...;Xn†… 5 : 40 †

ˆ



FY…y†ˆP…Yy†ˆP‰g…X†yŠ

ˆFX‰x:g…x†yŠ:

… 5 : 41 †

FX‰x:g…x†yŠˆ

Z



Z

…Rn:g…x†y†

fX…x†dx … 5 : 42 †



ˆ

FY…y†ˆ

Z

…R^2 :x 1 x 2 y†

Z

fX 1 X 2 …x 1 ;x 2 †dx 1 dx 2 : … 5 : 43 †

ˆ



argument x satisfies g( x) y.Intermsoff ( x), it is given by