Stat 5101 Lecture Notes - School of Statistics

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234 Stat 5101 (Geyer) Course NotesProof. This is just linearity of matrix multiplication.The second property means that all the eigenvectors corresponding to oneeigenvalue constitute a subspace. If the dimension of that subspace is k, thenit is possible to choose an orthonormal basis of k vectors that span the subspace.Since the first property of eigenvalues and eigenvectors says that (E.1)is also satisfied by eigenvectors corresponding to different eigenvalues, all of theeigenvectors chosen this way form an orthonormal set.Thus our orthonormal set of eigenvectors spans a subspace of dimension mwhich contains all eigenvectors of the matrix in question. The question thenarises whether this set is complete, that is, whether it is a basis for the wholespace, or in symbols whether m = n, where n is the dimension of the wholespace (A is an n × n matrix and the x i are vectors of dimension n). It turnsout that the set is always complete, and this is the third important fact abouteigenvalues and eigenvectors.Lemma E.3. Every real symmetric matrix has an orthonormal set of eigenvectorsthat form a basis for the space.In contrast to the first two facts, this is deep, and we shall not say anythingabout its proof, other than that about half of the typical linear algebra book isgiven over to building up to the proof of this one fact.The “third important fact” says that any vector can be written as a linearcombination of eigenvectorsn∑y = c i x ii=1and this allows a very simple description of the action of the linear operatordescribed by the matrixn∑n∑Ay = c i Ax i = c i λ i x ii=1i=1(E.4)So this says that when we use an orthonormal eigenvector basis, ifyhas therepresentation (c 1 ,...,c n ), then Ay has the representation (c 1 λ 1 ,...,c n λ n ).Let D be the representation in the orthonormal eigenvector basis of the linearoperator represented by A in the standard basis. Then our analysis above saysthe i-the element of Dc is c i λ i , that is,n∑d ij c j = λ i c i .j=1In order for this to hold for all real numbers c i , it must be that D is diagonald ii = λ id ij =0,i ≠ j
E.2. EIGENVALUES AND EIGENVECTORS 235In short, using the orthonormal eigenvector basis diagonalizes the linear operatorrepresented by the matrix in question.There is another way to describe this same fact without mentioning bases.Many people find it a simpler description, though its relation to eigenvalues andeigenvectors is hidden in the notation, no longer immediately apparent. Let Odenote the matrix whose columns are the orthonormal eigenvector basis (x 1 ,..., x n ), that is, if o ij are the elements of O, thenx i =(o 1i ,...,o ni ).Now (E.1) and (E.2) can be combined as one matrix equationO ′ O = I(E.5)(where, as usual, I is the n × n identity matrix). A matrix O satisfying thisproperty is said to be orthogonal. Another way to read (E.5) is that it saysO ′ = O −1 (an orthogonal matrix is one whose inverse is its transpose). Thefact that inverses are two-sided (AA −1 = A −1 A = I for any invertible matrixA) implies that OO ′ = I as well.Furthermore, the eigenvalue-eigenvector equation (E.3) can be written outwith explicit subscripts and summations asn∑n∑a ij o jk = λ k o ik = o ik d kk = o ij d jkj=1(where D is the the diagonal matrix with eigenvalues on the diagonal definedabove). Going back to matrix notation givesj=1AO = OD(E.6)The two equations (E.3) and (E.6) may not look much alike, but as we havejust seen, they say exactly the same thing in different notation. Using theorthogonality property (O ′ = O −1 ) we can rewrite (E.6) in two different ways.Theorem E.4 (Spectral Decomposition). Any real symmetric matrix Acan be writtenA = ODO ′(E.7)where D is diagonal and O is orthogonal.Conversely, for any real symmetric matrix A there exists an orthogonal matrixO such thatD = O ′ AOis diagonal.(The reason for the name of the theorem is that the set of eigenvalues issometimes called the spectrum of A). The spectral decomposition theorem saysnothing about eigenvalues and eigenvectors, but we know from the discussionabove that the diagonal elements of D are the eigenvalues of A, and the columnsof O are the corresponding eigenvectors.
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Stat 5101 Lecture NotesCharles J. G
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Contents1 Random Variables and Chan
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CONTENTSv5.1.4 Variance Matrices ..
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CONTENTSviiD Relations Among Brand
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Chapter 1Random Variables andChange
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1.1. RANDOM VARIABLES 3Sometimes it
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1.1. RANDOM VARIABLES 5that is, X i
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1.2. CHANGE OF VARIABLES 7the union
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1.2. CHANGE OF VARIABLES 9Thus even
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1.2. CHANGE OF VARIABLES 11Every in
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1.2. CHANGE OF VARIABLES 13where f
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1.3. RANDOM VECTORS 151.3.1 Discret
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1.4. THE SUPPORT OF A RANDOM VARIAB
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1.5. JOINT AND MARGINAL DISTRIBUTIO
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1.5. JOINT AND MARGINAL DISTRIBUTIO
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1.6. MULTIVARIABLE CHANGE OF VARIAB
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Chapter 2Expectation2.1 Introductio
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2.3. BASIC PROPERTIES 33expectation
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2.3. BASIC PROPERTIES 35that is, th
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2.4. MOMENTS 37The Multiplicativity
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2.4. MOMENTS 39holds for all real-v
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2.4. MOMENTS 41simple, it is often
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2.4. MOMENTS 43In contrast, for all
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2.4. MOMENTS 45is the sum of the ex
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2.4. MOMENTS 47Proof. Just take a i
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2.4. MOMENTS 49What happens to Coro
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2.4. MOMENTS 51This inequality is a
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2.4. MOMENTS 53(a)Why does (2.7) as
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2.5. PROBABILITY THEORY AS LINEAR A
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2.6. PROBABILITY IS A SPECIAL CASE
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2.7. INDEPENDENCE 772.7 Independenc
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2.7. INDEPENDENCE 79Then X and Y ar
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2.7. INDEPENDENCE 81Show that the f
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Chapter 3Conditional Probability an
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3.1. PARAMETRIC FAMILIES OF DISTRIB
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3.2. CONDITIONAL PROBABILITY DISTRI
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3.3. AXIOMS FOR CONDITIONAL EXPECTA
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3.4. JOINT, CONDITIONAL, AND MARGIN
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3.5. CONDITIONAL EXPECTATION AND PR
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Chapter 4Parametric Families ofDist
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4.1. LOCATION-SCALE FAMILIES 113The
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4.2. THE GAMMA DISTRIBUTION 115Show
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4.3. THE BETA DISTRIBUTION 117cours
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4.4. THE POISSON PROCESS 119Hence t
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4.4. THE POISSON PROCESS 121Definit
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4.4. THE POISSON PROCESS 123measure
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4.4. THE POISSON PROCESS 1254-3. A
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Chapter 5Multivariate DistributionT
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5.1. RANDOM VECTORS 129Again like r
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5.1. RANDOM VECTORS 1315.1.6 Covari
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5.1. RANDOM VECTORS 1335.1.7 Linear
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5.1. RANDOM VECTORS 135Example 5.1.
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5.1. RANDOM VECTORS 137Example 5.1.
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5.1. RANDOM VECTORS 139where we hav
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5.2. THE MULTIVARIATE NORMAL DISTRI
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5.3. BERNOULLI RANDOM VECTORS 151Th
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5.3. BERNOULLI RANDOM VECTORS 153yo
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5.4. THE MULTINOMIAL DISTRIBUTION 1
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Chapter 6Convergence Concepts6.1 Un
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6.1. UNIVARIATE THEORY 167It simpli
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6.1. UNIVARIATE THEORY 169density o
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6.1. UNIVARIATE THEORY 171Rewriting
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6.1. UNIVARIATE THEORY 173Comment T
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6.1. UNIVARIATE THEORY 175Problems6
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Chapter 7Sampling Theory7.1 Empiric
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7.1. EMPIRICAL DISTRIBUTIONS 1797.1
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7.1. EMPIRICAL DISTRIBUTIONS 181Def
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Page 219 and 220: Appendix AGreek LettersTable A.1: T
Page 221 and 222: Appendix BSummary of Brand-NameDist
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Page 239 and 240: Appendix EEigenvalues andEigenvecto
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Page 247: Appendix FNormal Approximations for
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