f G It expf fx.in xn xn IF Hat expf EE T vectorse Today we'll speed Multiua Reading 9 I 9.5 Last week we organized finite collectors of Us into

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1 Reading 9 I 9.5 Multiua random variables Last week we organized finite collectors of Us into vectors called rando vectorse Today we'll speed discussing Gaussian random vectors so fine Recall the univariate Gaussian PDF f G It expf and for a independent Gaussian RVs we have I fx.in xn xn IF e pf Hat expf EE T Since the Xi's are independent their covariance matrix is diagonal i

2 Define X EY xd c IR m Eu Milt The a it I i Hence expf EE I expftzlx mtc.ilx.us To write the La ti Tn term more cop atly we use two facts Fact Let A c IR with eigenvalus A In Then deffa II Ai Fact Let A E IR be a diagonal matrix Then the eigenvalues of A ore the diagonal elements of A Putting these together we see that IIT def Cx We can therefore write the joint PDF of the independent Gaussian RVs in matrix vector rotation as

3 expf Ek ult Ix ul fxkt det 12i.TK 1 The abore holds for independent RVs but what about in general It turns at the above PDF is the correct when density G 3 invertible ad when X satisfies the following A random vector X EX Xn is said to be Gaussian if i ie Cixi Ci EIR is a scalar Gaussian RV i.e every linear combination of its elements is a Gaussian If X has mean vector µ ad covariance C we write X Nfu c Ed X X independent Gaussias X is Gaussian E every sub vector of a Gaussian rudo vector is Gaussian Affintransformations Let Xu Nlm Cx 1 crr and be Rr What is the distribution of Y Axtb Is Y Gaussian First consider Z AX For Z to be Gaussian we need EZ to be a scalar Gaussian RV for every CelRr Note that

4 EZ c fax EA X but a A Las size lira and hence EZ is a linear conbination of the elements of X Therefore Z B Gaussian It is also easily checked that adding the constant a b to Z gives another Gaussian art hence Y AXtb is Gaussian What are its hee ad variance Elly EfAXtb AE b Arab EIENT Et Htt b HABIT Ef AXXTATt2AXbttbbt A RAT 2Ambt blot To get Cy we subtract EEYTEEYIT from EEYYT ETHIETY Ant b Amb Therefore y Ann AT 2Aybttbbt A RAT t 2Arbttbb Amrita't2Arbttbbt 1 Rx nut AT A GAT

5 Putting this all together we see that XnNfu G Y AXtb N Art b 1 At t UncorrelatedImpliesIndeperbity Jonty Gaussian RVs have the useful property that if they are uncorrelated they are independent Note tht in gene l we only hone the reverse implication Let your fu G The momentgeneratyfeti of X is M Is IEEet exp stut Estes where the second equality follows by taking Y six and applying the affine transformation formula above We now prove the fact stated above Let X Nfu G with uncorrelated elements i e Cx is diagonal with elements T fi Then texts exp simittsiri II exp II Mx Isi simittsiri

6 which is the MGF of an independent Gaussian RVs Since the MGF corresponds uniquely to the PDF we conclude that the PDF of X is that of a independent Gaussian RVs IconditionalExpecteteerandprobability Recall our study of MMSE estimation where we showed theorthogonality principle i.e that EEXIT satisfies Echl 4 0 X ELily for all fations ht For randomvectors the same property holds ELINT x EEXH where X YE IR Pope Let X Y be such that F is a Gaussian r d vector Then EExly Aly a ur g where A solves Acy Gy

7 Body We show that the proposed principle For simplicity assume rex o ad my tht the vector City f solution satisfies the orthogonality nasi o Now observe is Gaussian since it is a linear transformation of the Gaussianvector Let's look at the correlation between the top and bottom entries E IX Ayly'T E Exit 1 EEN'T Cy A Cy D y recall A solves AC Gy Hera KAY and Y are uncorrelated and therefore independent Therefore function for ht any E HY Ix Ay I 4411 ELI AY E htt O o completing the proof See pg 370 for the PDF of X ly

8 Probley Let y N o D and Y 3X Show that X and Y are jointly Gaussian ad find their c variance matrix Solution First we want to show that C Xt GY is Gaussian for arbitrary c ad c Note that C XtczY c Xt cz 3X K 3 a X RN 0, Since X Y are zero mean we have that cry XY and Cy y 4 Ext E Xl3X 3 Efx 3 E Y I EEx7 9 Therefore an f's

9 Problenteef Xi tn be RVs and define Yi i Xi k 1 a Suppose that Yi Ya are jointly Gaussian Determine whether X X are jointly Gaussian Solution Let's examine a few elements of Y to look for a pattern Y X Yz Y X 1 X X t X txz We can quickly see that Yu Yu tin or solving for Xu that Xu Yu Yu We can write this ie matrix form as X I 0 O O ti tt So X AY ad Y is Gaussian so X is as well A

10 Problem Let X Y U U be jointly Gaussian with X Y independent Nco 1 Let E det III If F ad T are uncorrelated find the conditional density fz.hu z uiu Solution First rate tht Z Xu Yu by evaluating the determinat Now consider the conditional CDF Fau zlu.ir PfZez U u.v u P XV yuez U u.v v P Xu Yuez U u.v u but I ad I are jointly Gaussian and uncorrelated ad therefore independent so we can dropthe conditioning to see that Fzwfzlu.ir P Xu Yu ez

11 Since X YinNlo D we have Therefore Xu Yu N o n'tu fzlu.ir Hun NN o n'tu

Multivariate Random Variable

Multivariate Random Variable Author: Author: Andrés Hincapié and Linyi Cao This Version: August 7, 2016 Multivariate Random Variable 3 Now we consider models with more than one r.v. These are called multivariate