Stat 5101 Lecture Notes - School of Statistics

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108 Stat 5101 (Geyer) Course NotesProblems3-1. In class we found the moment generating function of the geometric distribution(Section B.1.3 in Appendix B) is defined byψ(t) =1−p1−pe ton some neighborhood of zero. Find the variance of this random variable.3-2. Verify the details in (3.16a), (3.16b), and (3.16c).3-3. Suppose X is a positive random variable and the density of Y given X isf(y | x) = 2yx 2, 0
3.5. CONDITIONAL EXPECTATION AND PREDICTION 109(b) What is the marginal density of Y ?3-8. Suppose X and Z are independent random variables and E(Z) = 0. DefineY = X + X 2 + Z.(a)(b)(c)(d)Find E(Y | X).Find var(Y | X).What function of X is the best predictor of Y in the sense of minimizingexpected squared prediction error?What is the expected squared prediction error of this predictor?Note: Any of the answers may involve moments of X and Z.
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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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Page 87 and 88: 2.7. INDEPENDENCE 79Then X and Y ar
Page 89 and 90: 2.7. INDEPENDENCE 81Show that the f
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Page 97 and 98: 3.3. AXIOMS FOR CONDITIONAL EXPECTA
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Page 119 and 120: Chapter 4Parametric Families ofDist
Page 121 and 122: 4.1. LOCATION-SCALE FAMILIES 113The
Page 123 and 124: 4.2. THE GAMMA DISTRIBUTION 115Show
Page 125 and 126: 4.3. THE BETA DISTRIBUTION 117cours
Page 127 and 128: 4.4. THE POISSON PROCESS 119Hence t
Page 129 and 130: 4.4. THE POISSON PROCESS 121Definit
Page 131 and 132: 4.4. THE POISSON PROCESS 123measure
Page 133 and 134: 4.4. THE POISSON PROCESS 1254-3. A
Page 135 and 136: Chapter 5Multivariate DistributionT
Page 137 and 138: 5.1. RANDOM VECTORS 129Again like r
Page 139 and 140: 5.1. RANDOM VECTORS 1315.1.6 Covari
Page 141 and 142: 5.1. RANDOM VECTORS 1335.1.7 Linear
Page 143 and 144: 5.1. RANDOM VECTORS 135Example 5.1.
Page 145 and 146: 5.1. RANDOM VECTORS 137Example 5.1.
Page 147 and 148: 5.1. RANDOM VECTORS 139where we hav
Page 149 and 150: 5.2. THE MULTIVARIATE NORMAL DISTRI
Page 159 and 160: 5.3. BERNOULLI RANDOM VECTORS 151Th
Page 161 and 162: 5.3. BERNOULLI RANDOM VECTORS 153yo
Page 163 and 164: 5.4. THE MULTINOMIAL DISTRIBUTION 1
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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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7.1. EMPIRICAL DISTRIBUTIONS 183To
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7.2. SAMPLES AND POPULATIONS 185The
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7.2. SAMPLES AND POPULATIONS 187•
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7.3. SAMPLING DISTRIBUTIONS OF SAMP
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Appendix AGreek LettersTable A.1: T
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Appendix BSummary of Brand-NameDist
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B.1. DISCRETE DISTRIBUTIONS 215B.1.
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B.2. CONTINUOUS DISTRIBUTIONS 217Th
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B.2. CONTINUOUS DISTRIBUTIONS 219B.
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B.4. DISCRETE MULTIVARIATE DISTRIBU
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B.5. CONTINUOUS MULTIVARIATE DISTRI
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B.5. CONTINUOUS MULTIVARIATE DISTRI
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Appendix CAddition Rules forDistrib
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Appendix DRelations Among BrandName
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Appendix EEigenvalues andEigenvecto
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E.2. EIGENVALUES AND EIGENVECTORS 2
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E.2. EIGENVALUES AND EIGENVECTORS 2
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E.3. POSITIVE DEFINITE MATRICES 237
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Appendix FNormal Approximations for
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Stat 5101 Lecture Notes - School of Statistics

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