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Unformatted text preview: is time using
A= 11
01 det(A) = 1 A−1 = 1 −1
.
01 That is, the linear transformation used in this example maps u to α such that α = u + v and
β
v
β = v. The inverse mapping is given by u = α − β and v = β. or equivalently, u = A−1 α .
v
β
Proposition 4.7.1 yields:
fW,Z (α, β ) = fX,Y (α − β, β ) = fX (α − β )fY (β ),
for all (α, β ) ∈ R2 . The marginal pdf of W is obtained by integrating out β :
−∞ fX (α − β )fY (β )dβ. fW (α) =
∞ Equivalently, fW is the convolution: fW = fX ∗ fY . This expression for the pdf of the sum of two
independent continuoustype random variables was found by a diﬀerent method in Section 4.5.2. 4.7.2 Transformation of pdfs under a onetoone mapping We shall discuss next how joint pdfs are transformed for possibly nonlinear functions. Speciﬁcally,
we assume that W = g ( X ), where g is a mapping from R2 to R2 . As in the special case of
Z
Y
linear transformations, think of the mapping as going from the u − v plane to the α − β plane.
Therefore, for each (u, v ), there corre...
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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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