Isye 2027

# Ac ab complement of a ab a b any element of a is

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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.

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