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Unformatted text preview: Proof. It suffices to prove the proposition in case µX = µY = 0 and σX = σY = 1, because these parameters simply involve centering and scaling of X and Y separately, or, equivalently, translation and scaling of the joint pdf parallel to the u-axis or parallel to the v -axis. Such centering and scaling of X and Y separately does not change the correlation coefficient, as shown in Section 4.8. 4.11. JOINT GAUSSIAN DISTRIBUTION 171 The joint pdf in this case can be written as the product of two factors, as follows: fX,Y (u, v ) = = 1 2π 1 − ρ2 exp − u2 + v 2 − 2ρuv 2(1 − ρ2 ) 1 u2 √ exp − 2 2π 1 2π (1 − ρ2 ) exp − (v − ρu)2 2(1 − ρ2 ) . (4.38) The first factor is a function of u alone, and is the standard normal pdf. The second factor, as a function of v for u fixed, is a Gaussian pdf with mean ρu and variance 1 − ρ2 . In particular, the integral of the second factor with respect to v is one. Therefore, the first factor is the marginal pdf of X and 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 Tech.

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