Transcription of Multivariate Distributions - CMU Statistics
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Multivariate Distributions - CMU Statistics
23:15 Wednesday 27thFebruary, 2013 chapter 14 Multivariate Review of DefinitionsLet s review some definitions from basic probability . When we have a random vector~Xwithpdifferent components,X1,X2,..Xp, thejoint cumulative distributionfunctionisF(~a)=F(a1,a2,..ap )=Pr X1 a1,X2 a2,..Xp ap ( )ThusF(~b) F(~a)=Pr a1<X1 b1,a2<X2 b2,..ap<Xp bp ( )This is the probability thatXis in a (hyper-)rectangle, rather than just in an probability density @ap ~a=~x( )Of course,F(~a)=Za1 1Za2 1p(x1,x2,..xp) ( )(In this case, the order of integration doesn t matter. Why?)From these, and especially from the joint PDF, we can recover the marginal PDFof any group of variables, say those numbered 1 throughq,p(x1,x2.)
the probability density of the multivariate Gaussian is p ... 14.3 Inference with Multivariate Distributions ... parametric inference is covered in Chapter 15. 14.3.1 Estimation The oldest method of estimating parametric distributions is moment-matching or the method of moments. If there are q unknown parameters of the distribution,
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