Stat 5101 Lecture Notes - School of Statistics

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ivStat 5101 (Geyer) Course Notes2.5 Probability Theory as Linear Algebra ............... 552.5.1 The Vector Space L 1 ..................... 562.5.2 Two Notions of Linear Functions .............. 582.5.3 Expectation on Finite Sample Spaces ........... 592.5.4 Axioms for Expectation (Part II) .............. 622.5.5 General Discrete Probability Models ............ 642.5.6 Continuous Probability Models ............... 662.5.7 The Trick of Recognizing a Probability Density . ..... 682.5.8 Probability Zero ....................... 682.5.9 How to Tell When Expectations Exist ........... 702.5.10 L p Spaces ........................... 742.6 Probability is a Special Case of Expectation ............ 752.7 Independence . ............................ 772.7.1 Two Definitions ........................ 772.7.2 The Factorization Criterion ................. 772.7.3 Independence and Correlation ................ 783 Conditional Probability and Expectation 833.1 Parametric Families of Distributions ................ 833.2 Conditional Probability Distributions ................ 863.3 Axioms for Conditional Expectation ................ 883.3.1 Functions of Conditioning Variables ............ 883.3.2 The Regression Function ................... 893.3.3 Iterated Expectations .................... 913.4 Joint, Conditional, and Marginal .................. 953.4.1 Joint Equals Conditional Times Marginal ......... 953.4.2 Normalization ......................... 973.4.3 Renormalization ....................... 983.4.4 Renormalization, Part II ...................1013.4.5 Bayes Rule ..........................1033.5 Conditional Expectation and Prediction ..............1054 Parametric Families of Distributions 1114.1 Location-Scale Families .......................1114.2 The Gamma Distribution ......................1154.3 The Beta Distribution ........................1174.4 The Poisson Process .........................1194.4.1 Spatial Point Processes ...................1194.4.2 The Poisson Process .....................1204.4.3 One-Dimensional Poisson Processes .............1225 Multivariate Theory 1275.1 Random Vectors ...........................1275.1.1 Vectors, Scalars, and Matrices ................1275.1.2 Random Vectors .......................1285.1.3 Random Matrices .......................128
CONTENTSv5.1.4 Variance Matrices ......................1295.1.5 What is the Variance of a Random Matrix? ........1305.1.6 Covariance Matrices .....................1315.1.7 Linear Transformations ...................1335.1.8 Characterization of Variance Matrices ...........1355.1.9 Degenerate Random Vectors . ...............1365.1.10 Correlation Matrices .....................1405.2 The Multivariate Normal Distribution ...............1415.2.1 The Density ..........................1435.2.2 Marginals ...........................1465.2.3 Partitioned Matrices .....................1465.2.4 Conditionals and Independence ...............1485.3 Bernoulli Random Vectors ......................1515.3.1 Categorical Random Variables ...............1525.3.2 Moments ...........................1535.4 The Multinomial Distribution ....................1545.4.1 Categorical Random Variables ...............1545.4.2 Moments ...........................1555.4.3 Degeneracy ..........................1555.4.4 Density ............................1565.4.5 Marginals and “Sort Of” Marginals ............1575.4.6 Conditionals .........................1596 Convergence Concepts 1656.1 Univariate Theory ..........................1656.1.1 Convergence in Distribution . . ...............1656.1.2 The Central Limit Theorem . . ...............1666.1.3 Convergence in Probability . . ...............1696.1.4 The Law of Large Numbers . . ...............1706.1.5 The Continuous Mapping Theorem .............1706.1.6 Slutsky’s Theorem ......................1716.1.7 Comparison of the LLN and the CLT ...........1726.1.8 Applying the CLT to Addition Rules ............1726.1.9 The Cauchy Distribution . . . ...............1747 Sampling Theory 1777.1 Empirical Distributions .......................1777.1.1 The Mean of the Empirical Distribution ..........1797.1.2 The Variance of the Empirical Distribution ........1797.1.3 Characterization of the Mean . ...............1807.1.4 Review of Quantiles .....................1807.1.5 Quantiles of the Empirical Distribution ..........1817.1.6 The Empirical Median ....................1837.1.7 Characterization of the Median ...............1837.2 Samples and Populations ......................1857.2.1 Finite Population Sampling . . ...............185
Page 1 and 2: Stat 5101 Lecture NotesCharles J. G
Page 3: Contents1 Random Variables and Chan
Page 7 and 8: CONTENTSviiD Relations Among Brand
Page 9 and 10: Chapter 1Random Variables andChange
Page 11 and 12: 1.1. RANDOM VARIABLES 3Sometimes it
Page 13 and 14: 1.1. RANDOM VARIABLES 5that is, X i
Page 15 and 16: 1.2. CHANGE OF VARIABLES 7the union
Page 17 and 18: 1.2. CHANGE OF VARIABLES 9Thus even
Page 19 and 20: 1.2. CHANGE OF VARIABLES 11Every in
Page 21 and 22: 1.2. CHANGE OF VARIABLES 13where f
Page 23 and 24: 1.3. RANDOM VECTORS 151.3.1 Discret
Page 25 and 26: 1.4. THE SUPPORT OF A RANDOM VARIAB
Page 27 and 28: 1.5. JOINT AND MARGINAL DISTRIBUTIO
Page 29 and 30: 1.5. JOINT AND MARGINAL DISTRIBUTIO
Page 31 and 32: 1.6. MULTIVARIABLE CHANGE OF VARIAB
Page 39 and 40: Chapter 2Expectation2.1 Introductio
Page 41 and 42: 2.3. BASIC PROPERTIES 33expectation
Page 43 and 44: 2.3. BASIC PROPERTIES 35that is, th
Page 45 and 46: 2.4. MOMENTS 37The Multiplicativity
Page 47 and 48: 2.4. MOMENTS 39holds for all real-v
Page 49 and 50: 2.4. MOMENTS 41simple, it is often
Page 51 and 52: 2.4. MOMENTS 43In contrast, for all
Page 53 and 54: 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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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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