Stat 5101 Lecture Notes - School of Statistics

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184 Stat 5101 (Geyer) Course NotesProof. What we need to show is that if m is a median, that is, ifand a is any real number, thenP (X ≤ m) ≥ 1 2and P (X ≥ m) ≥ 1 2E(|X − a|) ≥ E(|X − m|).Without loss of generality, we may suppose a>m. (The case a = m is trivial.The case a
7.2. SAMPLES AND POPULATIONS 185There is no end to this game. Every notion that is defined for generalprobability models, we can specialize to empirical distributions. We can defineempirical moments and central moments of all orders, and so forth and so on.But we won’t do that in gory detail. What we’ve done so far is enough for now.7.2 Samples and Populations7.2.1 Finite Population SamplingIt is common to apply statistics to a sample from a population. The populationcan be any finite set of individuals. Examples are the population ofMinnesota today, the set of registered voters in Minneapolis on election day, theset of wolves in Minnesota. A sample is any subset of the population. Examplesare the set of voters called by an opinion poll and asked how they intendto vote, the set of wolves fitted with radio collars for a biological experiment.By convention we denote the population size by N and the sample size by n.Typically n is much less than N. For an opinion poll, n is typically about athousand, and N is in the millions.Random SamplingA random sample is one drawn so that every individual in the population isequally likely to be in the sample. There are two kinds.Sampling without Replacement The model for sampling without replacementis dealing from a well-shuffled deck of cards. If we deal n cards from adeck of N cards, there are (N) n possible outcomes, all equally likely (here weare considering that the order in which the cards are dealt matters). Similarlythere are (N) n possible samples without replacement of size n from a populationof size N. If the samples are drawn in such a way that all are equally likely wesay we have a random sample without replacement from the population.Sampling with Replacement The model for sampling with replacement isspinning a roulette wheel. If we do n spins and the wheel has N pockets, thereare N n possible outcomes, all equally likely. Similarly there are N n possiblesamples with replacement of size n from a population of size N. If the samplesare drawn in such a way that all are equally likely we say we have a randomsample with replacement from the population.Lindgren calls this a simple random sample, although there is no standardmeaning of the word “simple” here. Many statisticians would apply “simple” tosampling either with or without replacement using it to mean that all samplesare equally likely in contrast to more complicated sampling schemes in whichthe samples are not all equally likely.
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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
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
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Page 159 and 160: 5.3. BERNOULLI RANDOM VECTORS 151Th
Page 161 and 162: 5.3. BERNOULLI RANDOM VECTORS 153yo
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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: 7.1. EMPIRICAL DISTRIBUTIONS 183To
Page 195 and 196: 7.2. SAMPLES AND POPULATIONS 187•
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 and 224: B.1. DISCRETE DISTRIBUTIONS 215B.1.
Page 225 and 226: B.2. CONTINUOUS DISTRIBUTIONS 217Th
Page 227 and 228: B.2. CONTINUOUS DISTRIBUTIONS 219B.
Page 229 and 230: B.4. DISCRETE MULTIVARIATE DISTRIBU
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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
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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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