Stat 5101 Lecture Notes - School of Statistics

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216 Stat 5101 (Geyer) Course NotesThe DensityMomentsf(x) =( ) x−1p k (1 − p) x−k , x = k, k +1,....k−1E(X) = k pk(1 − p)var(X)=p 2SpecializationGeo(p) = NegBin(1,p)B.1.7The Negative Binomial Distribution, Type IIThe Abbreviation No abbreviation to avoid confusion with the other typedefined in Section B.1.6.Relation Between the Types If X ∼ NegBin(k, p), then Y = X − k hasthe distribution defined in this section.X is the number of trials before the k-th success in an i. i. d. sequence ofBer(p) random variables. Y is the number of failures before the k-th success.The Sample Space The integers 0, 1, ....The Parameterp such that 0
B.2. CONTINUOUS DISTRIBUTIONS 217The ParameterThe Densityp such that 0
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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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Page 173 and 174: Chapter 6Convergence Concepts6.1 Un
Page 175 and 176: 6.1. UNIVARIATE THEORY 167It simpli
Page 177 and 178: 6.1. UNIVARIATE THEORY 169density o
Page 179 and 180: 6.1. UNIVARIATE THEORY 171Rewriting
Page 181 and 182: 6.1. UNIVARIATE THEORY 173Comment T
Page 183 and 184: 6.1. UNIVARIATE THEORY 175Problems6
Page 185 and 186: Chapter 7Sampling Theory7.1 Empiric
Page 187 and 188: 7.1. EMPIRICAL DISTRIBUTIONS 1797.1
Page 189 and 190: 7.1. EMPIRICAL DISTRIBUTIONS 181Def
Page 191 and 192: 7.1. EMPIRICAL DISTRIBUTIONS 183To
Page 193 and 194: 7.2. SAMPLES AND POPULATIONS 185The
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Page 197 and 198: 7.3. SAMPLING DISTRIBUTIONS OF SAMP
Page 219 and 220: Appendix AGreek LettersTable A.1: T
Page 221 and 222: Appendix BSummary of Brand-NameDist
Page 223: B.1. DISCRETE DISTRIBUTIONS 215B.1.
Page 227 and 228: B.2. CONTINUOUS DISTRIBUTIONS 219B.
Page 229 and 230: B.4. DISCRETE MULTIVARIATE DISTRIBU
Page 231 and 232: B.5. CONTINUOUS MULTIVARIATE DISTRI
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Page 235 and 236: Appendix CAddition Rules forDistrib
Page 237 and 238: Appendix DRelations Among BrandName
Page 239 and 240: Appendix EEigenvalues andEigenvecto
Page 241 and 242: E.2. EIGENVALUES AND EIGENVECTORS 2
Page 243 and 244: E.2. EIGENVALUES AND EIGENVECTORS 2
Page 245 and 246: E.3. POSITIVE DEFINITE MATRICES 237
Page 247: Appendix FNormal Approximations for
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Stat 5101 Lecture Notes - School of Statistics

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