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Unformatted text preview: eimage of S under the linear transformation. Since
S is a small set, R is also a small set. So P {(W, Z ) ∈ S } = P {(X, Y ) ∈ R} ≈ fX,Y (u, v )area(R).
)
1
Thus, fW,Z (α.β ) ≈ fX,Y (x, y ) area(R) =  det A fX,Y (x, y ). This observation leads to the following
area(S
proposition:
Proposition 4.7.1 Suppose W = A X , where
Z
Y
det(A) = 0. Then W has joint pdf given by
Z
fW,Z (α, β ) = X
Y 1
fX,Y
 det A has pdf fX,Y , and A is a matrix with A−1 α
β . Example 4.7.2 Suppose X and Y have joint pdf fX,Y , and W = X − Y and Z = X + Y. Express
the joint pdf of W and Z in terms of fX,Y . 146 CHAPTER 4. JOINTLY DISTRIBUTED RANDOM VARIABLES Solution: We apply Proposition 4.7.1, using
A= 1 −1
11 det(A) = 2 1
2 A−1 = 1
−2 1
2
1
2 . u
α
v to β such that α = u
−α+β
u
2 , or equivalently, v = That is, the linear transformation used in this example maps
α+β
2 β = u + v. The inverse mapping is given by u =
Proposition 4.7.1 yields:
1
fW,Z (α, β ) = fX,Y
2 and v = α + β −α + β
,
2
2 − v and
A−1 α
β . , for all (α, β ) ∈ R2 . Example 4.7.3 Suppose X and Y are independent, continuoustype random variables. Find the
joint pdf of W and Z, where W = X + Y and Z = Y. Also, ﬁnd the pdf of W.
Solution: We again apply Proposition 4.7.1, th...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Zahrn
 The Land

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