Jointly distributed random variables for constants a

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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 continuous-type random variables was found by a diﬀerent method in Section 4.5.2. 4.7.2 Transformation of pdfs under a one-to-one 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.

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