Stat 5101 Lecture Notes - School of Statistics

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118 Stat 5101 (Geyer) Course Notesis an unnormalized probability density. This is clear when s ≥ 1 and t ≥ 1,because then it is bounded. When s
4.4. THE POISSON PROCESS 119Hence the Jacobian isJ(u, v) =∣∂x∂u∂y∂u∂x∂v∂y∂v∣ ∣∣∣∣∣ ∣ = v u1−v −u∣ = −uThe joint density of X and Y is f X (x)f Y (y) by independence. By the changeof variable formula, the joint density of U and V isf U,V (u, v) =f X,Y [uv, u(1 − v)]|J(u, v)|= f X (uv)f Y [u(1 − v)]u= λsλtΓ(s) (uv)s−1 e −λuvΓ(t) [u(1 − v)]t−1 e −λu(1−v) u= λs+tΓ(s)Γ(t) us+t−1 e −λu v s−1 (1 − v) t−1Since the joint density factors into a function of u times a function of v, thevariables U and V are independent. Since these functions are proportional tothe gamma and beta densities asserted by the theorem, U and V must actuallyhave these distributions.Corollary 4.3.B(s, t) = Γ(s)Γ(t)Γ(s + t)Proof. The constant in the joint density found in the proof of the theorem mustbe the product of the constants for the beta and gamma densities. Henceλ s+tΓ(s)Γ(t) =Solving for B(s, t) gives the corollary.λs+t 1Γ(s + t) B(s, t)For moments of the beta distribution, see Lindgren pp. 176–177.4.4 The Poisson Process4.4.1 Spatial Point ProcessesA spatial point process is a random pattern of points in a region of space.The space can be any dimension.A point process is simple if it never has points on top of each other so thateach point of the process is at a different location in space. A point process isboundedly finite if with probability one it has only a finite number of points inany bounded set.Let N A denote the number of points in a region A. Since the point patternis random, N A is a random variable. Since it counts points, N A is a discrete
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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 75 and 76: 2.5. PROBABILITY THEORY AS LINEAR A
Page 83 and 84: 2.6. PROBABILITY IS A SPECIAL CASE
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
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Page 95 and 96: 3.2. CONDITIONAL PROBABILITY DISTRI
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Page 103 and 104: 3.4. JOINT, CONDITIONAL, AND MARGIN
Page 113 and 114: 3.5. CONDITIONAL EXPECTATION AND PR
Page 119 and 120: Chapter 4Parametric Families ofDist
Page 121 and 122: 4.1. LOCATION-SCALE FAMILIES 113The
Page 123 and 124: 4.2. THE GAMMA DISTRIBUTION 115Show
Page 125: 4.3. THE BETA DISTRIBUTION 117cours
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
Page 173 and 174: Chapter 6Convergence Concepts6.1 Un
Page 175 and 176: 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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