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Unformatted text preview: e second factor is the conditional pdf of Y given X = u : 1 u2 √ exp − 2 2π fX (u) = 1 fY |X (v |u) = 2π (1 − ρ2 ) exp − (v − ρu)2 2(1 − ρ2 ) . (4.39) Thus, X is a standard normal random variable. By symmetry, Y is also a standard normal random variable. This proves (a). The class of bivariate normal pdfs is preserved under linear transformations corresponding to multiplication of X by a matrix A if det A = 0. Given a and b, we can select c and d so that Y b the matrix A = a d has det(A) = ad − bc = 0. Then the random vector A X has a bivariate Y c normal pdf, so by part (a) already proven, both of its coordinates are Gaussian random variables. In particular, its first coordinate, aX + bY, is a Gaussian random variable. This proves (b). By (4.39), given X = u, the conditional distribution of Y is Gaussian with mean ρu and variance 1 − ρ2 . Therefore, g ∗ (u) = E [Y |X = u] = ρu. Since X and Y are both standard (i.e. they have mean zero and variance one), ρX,Y...
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