Stat 5101 Lecture Notes - School of Statistics

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228 Stat 5101 (Geyer) Course NotesPoissonIf X 1 , ..., X k are independent with X i ∼ Poi(µ i ), thenX 1 + ···+X k ∼Poi(µ 1 + ···+µ k ).(C.5)ExponentialIf X 1 , ..., X k are i. i. d. Exp(λ), thenX 1 + ···+X k ∼Gam(n, λ).(C.6)• All the exponential distributions must have the same rate parameter λ.GammaIf X 1 , ..., X k are independent with X i ∼ Gam(α i ,λ), thenX 1 + ···+X k ∼Gam(α 1 + ···+α k ,λ).(C.7)• All the gamma distributions must have the same rate parameter λ.• (C.6) is the special case of (C.7) obtained by setting α 1 = ···=α k =1.Chi-SquareIf X 1 , ..., X k are independent with X i ∼ chi 2 (n i ), thenX 1 + ···+X k ∼chi 2 (n 1 + ···+n k ).(C.8)• (C.8) is the special case of (C.7) obtained by settingα i = n i /2 and λ i =1/2, i =1,...,k.NormalIf X 1 , ..., X k are independent with X i ∼N(µ i ,σi 2 ), thenX 1 + ···+X k ∼N(µ 1 +···+µ k ,σ 2 1 +···+σ 2 k).(C.9)Linear Combination of Normals If X 1 , ...,X k are independent with X i ∼N (µ i ,σi 2) and a 1, ..., a k are constants, then(k∑k∑a i X i ∼N a i µ i ,i=1i=1)k∑a 2 i σi2 . (C.10)• (C.9) is the special case of (C.10) obtained by setting a 1 = ···=a k =1.i=1CauchyIf X 1 , ..., X k are independent with X i ∼ Cauchy(µ, σ), thenX 1 + ···+X k ∼Cauchy(nµ, nσ).(C.11)
Appendix DRelations Among BrandName DistributionsD.1 Special CasesFirst there are the special cases, which were also noted in Appendix B.Ber(p) = Bin(1,p)Geo(p) = NegBin(1,p)Exp(λ) = Gam(1,λ)chi 2 (k) = Gam ( k2 , )12The main point of this appendix are the relationships that involve moretheoretical issues.D.2 Relations Involving Bernoulli SequencesSuppose X 1 , X 2 , ... are i. i. d. Ber(p) random variables.If n is a positive integer andY = X 1 + ···+X nis the number of “successes” in the n Bernoulli trials, thenY ∼ Bin(n, p).On the other hand, if y is positive integer and N is the trial at which they-th success occurs, that is the random number N such thatX 1 + ···+X N =yX 1 +···+X k
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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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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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Page 219 and 220: Appendix AGreek LettersTable A.1: T
Page 221 and 222: Appendix BSummary of Brand-NameDist
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Page 247: Appendix FNormal Approximations for
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