Stat 5101 Lecture Notes - School of Statistics

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220 Stat 5101 (Geyer) Course NotesThe Sample SpaceThe interval (0, ∞) of the real numbers.The Parameter A positive integer k.The DensityMomentsf(x) =12 k/2 Γ(k/2) xk/2−1 e −x/2 , x > 0.E(X) =kvar(X) =2kGeneralizationchi 2 (k) = Gam ( k2 , )12B.2.7The Cauchy DistributionThe AbbreviationCauchy(µ, σ).The Sample Space The real line R.The Parametersµ and σ such that σ>0.The Densityf(x) = 1 π·σσ 2 +(x−µ) 2, x ∈ R.MomentsNone: E(|X|) =∞.B.3 Special FunctionsB.3.1The Gamma FunctionThe DefinitionΓ(α) =∫ ∞0x α−1 e −x dx, α > 0 (B.2)The Recursion RelationΓ(α +1)=αΓ(α)(B.3)
B.4. DISCRETE MULTIVARIATE DISTRIBUTIONS 221Known ValuesΓ(1) = 1and hence using the recursion relationΓ(n +1)=n!for any nonnegative integer n.AlsoΓ( 1 2 )=√ πand hence using the recursion relationand so forth.Γ( 3 2 )= 1 2√ πΓ( 5 2 )= 3 ·12 2√ πΓ( 7 2 )= 5 ·3 ·12 2 2√ πB.3.2The Beta FunctionThe function B(s, t) defined by (B.1).B.4 Discrete Multivariate DistributionsB.4.1The Multinomial DistributionThe Abbreviationfrom context.Multi k (n, p) or Multi(n, p) if the dimension k is clearThe Sample SpaceS = { y ∈ N k : y 1 + ···y k =n}where N denotes the “natural numbers” 0, 1, 2, ....The Parameter p =(p 1 ,...,p k ) such that p i ≥ 0 for all i and ∑ i p i =1.The Density( )n ∏ kf(y) =y 1 ,...,y kj=1p yjj ,y ∈ S
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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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Chapter 6Convergence Concepts6.1 Un
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6.1. UNIVARIATE THEORY 167It simpli
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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
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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 and 224: B.1. DISCRETE DISTRIBUTIONS 215B.1.
Page 225 and 226: B.2. CONTINUOUS DISTRIBUTIONS 217Th
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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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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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