Stat 5101 Lecture Notes - School of Statistics

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112 Stat 5101 (Geyer) Course NotesIt is clear that as µ and σ run over all possible values we get the same familyof distributions as before. The parameter values that go with each particulardistribution have changed, but each density that appears in one family alsoappears in the other. The correspondence between the parameters in the twoparameterizations isµ ←→ µ + σασ ←→ σβIf the reference random variable X has a variance, then every distribution inthe family has a variance (by Theorem 2.44 in these notes), and the distributionsof the family have every possible mean and variance. Since we are free to choosethe reference distribution as any distribution in the family, we may as well chooseso that E(X) = 0 and var(X) = 1, then µ is the mean and σ the standarddeviation of the variable Y with density f µ,σ .But the distributions of the family do not have to have either means or variances.In that case we cannot call µ the mean or σ the standard deviation. Thatis the reason why in general we call µ and σ the location and scale parameters.Example 4.1.1 (Uniform Distributions).The U(a, b) family of distribution defined in Section B.2.1 of Appendix B hasdensitiesf(x | a, b) = 1 , a
4.1. LOCATION-SCALE FAMILIES 113Then use the formula for a general location-scale family, obtainingf(y | µ, σ) = 1 ( ) y−µσ f = 1σ 2σ √ 3on the domain of definition, whatever that is. The change of variable is y =µ+σx, sox=± √ 3 maps to µ±σ √ 3, and those are the endpoints of the domainof definition. Sof(y | µ, σ) = 12σ √ 3 , µ−σ√ 3
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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 85 and 86: 2.7. INDEPENDENCE 772.7 Independenc
Page 87 and 88: 2.7. INDEPENDENCE 79Then X and Y ar
Page 89 and 90: 2.7. INDEPENDENCE 81Show that the f
Page 91 and 92: Chapter 3Conditional Probability an
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Page 95 and 96: 3.2. CONDITIONAL PROBABILITY DISTRI
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Page 113 and 114: 3.5. CONDITIONAL EXPECTATION AND PR
Page 119: Chapter 4Parametric Families ofDist
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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