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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 ) =
2π 1 − ρ2 exp − u2 + v 2 − 2ρuv
2(1 − ρ2 ) 1
√ exp −
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.
- Spring '08
- The Land