Gaussian Random Vectors - Math

Gaussian Random Vectors 1. The multivariate normal distribution Let X := (X1 X ) be a Random vector . We say that X is a Gaussian Random vector if we can write X = + AZ . where R , A is an matrix and Z := (Z1 Z ) is a - vector of standard normal Random variables. Proposition 1. Let X be a Gaussian Random vector , as above. Then, 1 2 1 . EX = Var(X) := = AA and MX ( ) = e + 2 A = e + 2 . for all R . Thanks to the uniqueness theorem of MGF's it follows that the dis- tribution of X is determined by , , and the fact that it is multivariate normal. From now on, we sometimes write X N ( ), when we mean that MX ( ) = exp( + 21 ). Interesetingly enough, the choice of A and Z are typically not unique; only ( ) influences the distribu- tion of X. Proof. The expectation of X is , since E(AZ) = AE(Z) = 0. Also, . E(XX ) = E [ + AZ] [ + AZ] = + AE(ZZ )A . Since E(ZZ ) = I, the variance-covariance of X is E(XX ) (EX)(EX) =. E(XX ) = AA , as desired. Finally, note that MX ( ) = exp( ).

Gaussian Random Vectors 1. The multivariate normal distribution Let X:= (X1 ￿￿￿￿￿X￿)￿ be a random vector. We say that X is a Gaussian random vector if we can write X = µ +AZ￿ where µ ∈ R￿, A is an ￿ × ￿ matrix and Z:= (Z1 ￿￿￿￿￿Z￿)￿ is a ￿-vector of i.i.d. standard normal random variables. Proposition 1.

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