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Unformatted text preview: f fX,Y (u, v ).
Suppose X and Y have a joint pdf fX,Y , and suppose W = aX + bY and Z = cX + dY for
some constants a, b, c, and d. Equivalently, in matrix notation, suppose X has a joint pdf fX,Y ,
Y
and suppose
W
X
ab
=A
where A =
.
Z
Y
cd
Thus, we begin with a random point X and get another random point W . For ease of analysis,
Y
Z
we can suppose that X is in the u − v plane and W is in the α − β plane. That is, W is the
Y
Z
Z
image of X under the linear mapping:
Y
α
β =A u
.
v The determinant of A is deﬁned by det(A) = ad − bc. If det A = 0 then the mapping has an inverse,
given by
1
α
u
d −b
where A−1 =
= A− 1
.
β
v
det A −c a
An important property of such linear transformations is that if R is a set in the u − v plane and
if S is the image of the set under the mapping, then area(S ) =  det(A)area(R), where det(A) is
the determinant of A. Consider the problem of ﬁnding fW,Z (α, β ) for some ﬁxed (α, β ). If there is
∈
a small rectangle S with a corner at (α, β ), then fW,Z (α, β ) ≈ P {(W,Z()S )S } . Now {(W, Z ) ∈ S } is
area
the same as {(X, Y ) ∈ R}, where R is the pr...
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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 Tech.
 Spring '08
 Zahrn
 The Land

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