STAT 730 Chapter 3: Normal Distribution Theory
STAT 730 chapter 3 : Normal Distribution TheoryTimothy HansonDepartment of Statistics, University of South CarolinaStat 730: multivariate Analysis1 / 36Nice properties of multivariate Normal random vectorsMultivariate Normal easily generalizes univariate harder to generalize Poisson, gamma, exponential, completely by first and second moments, meanvector and covariance Np( , ), then ij= 0 impliesxiindependent x N(a ,a a).Central Limit Theorem says sample means are approximatelymultivariate geometry makes properties / 36Definition via Cram er-Woldxis multivariate Normal a xis Normal for nx Np( , ) a x N(a ,a a) for alla : Ifx Np( , ) then its characteristic function is x(t) = exp(it 12t t).Proof: Lety=t x. Then the ofyis y(s)def=E{eisy}= exp{isE(y) 12s2var(y)}= exp{ist 12s2t t}.Then the ofxis x(t)def=E{eit x}= y(1) = exp(it 12t t).
STAT 730 Chapter 3: Normal Distribution Theory Timothy Hanson DepartmentofStatistics,UniversityofSouthCarolina Stat730: MultivariateAnalysis 1/36. Nice properties of multivariate normal random vectors Multivariate normal easily generalizes univariate normal. Much harder to generalize Poisson, gamma, exponential, etc. ... (Chapter 2). ...
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