Unformatted text preview: iables, by the CLT. But also for any constants a and b, aX + bY
···+(
must be Gaussian, because aX + bY has the limiting distribution of (aU1 +bV1 )+√n aUn +bVn ) , which
is Gaussian by the CLT. This observation motivates the following deﬁnition:
Deﬁnition 4.11.1 Random variables X and Y are said to be jointly Gaussian if every linear
combination aX + bY is a Gaussian random variable. (For the purposes of this deﬁnition, a
constant is considered to be a Gaussian random variable with variance zero.)
Being jointly Gaussian includes the case that X and Y are each Gaussian and linearly related:
X = aY + b for some a, b or Y = aX + b for some a, b. In these cases, X and Y do not have a
joint pdf. Aside from those two degenerate cases, a pair of jointly Gaussian random variables has
a bivariate normal (or Gaussian) pdf, given by 2
2
u−µX
µ
µ
v −µY
+ v−Y Y
− 2ρ u−XX
σX
σ
σ
σY
1 fX,Y (u, v ) =
exp −
(4.37)
,
2)
2
2(1 − ρ
2πσX σY 1 − ρ
where the ﬁve pa...
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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
 Zahrn
 The Land

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