Stat 5101 Lecture Notes - School of Statistics

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100 Stat 5101 (Geyer) Course Notesfor some normalizing constant c. Plugging this into (3.21) givesf(y | x) =h(x, y)∫h(x, y) dy(3.22)The c’s cancel in the numerator and denominator.Our slogan about conditional probability and renormalization helps us rememberwhich marginal is meant inconditional =jointmarginal• If the conditional in question is f(y | x), then we are considering y thevariable (x is fixed).• Thus the marginal in question is the one obtained by integrating withrespect to y (that’s what we are considering variable).• The marginal obtained by integrating out y is the marginal of the othervariable (slogan on p. 19 in these notes). Hence the marginal is f X (x).But even if you are confused about how to calculate marginals or whichmarginal you need to divide by, you should still be able to calculate conditionalsusing (3.21) and (3.22), which contain no marginals and are in fact derivableon the spot. Both are obvious consequences of the facts that• Conditional densities are proportional to joint densities considered as functionsof the variable(s) in front of the bar.• Conditional densities integrate to one considered as functions of the variable(s)in front of the bar.Example 3.4.3.Consider the functionh(x, y) =(x+y 2 )e −x−y , x > 0, y > 0.If we take this to be an unnormalized joint density, what are the two conditionaldensities f(x | y) and f(y | x)?Integrating with respect to x gives∫ ∞∫ ∞∫ ∞h(x, y) dx = e −y xe −x dx + y 2 e −y e −x dx00=(1+y 2 )e −y0We used the formula∫ ∞0x n e −x dx =Γ(n+1)=n! (3.23)
3.4. JOINT, CONDITIONAL, AND MARGINAL 101to evaluate the integrals. HenceSimilarly∫ ∞0f(x | y) =f(x, y)∫f(x, y) dx= x + y21+y 2e−x∫ ∞∫ ∞h(x, y) dy = xe −x e −y dy + e −x y 2 e −y dy0=(x+2)e −xAgain, we used (3.23) to evaluate the integrals. Sof(y | x) =f(x, y)∫f(x, y) dy= x + y2x +2 e−yThings become considerably more complicated when the support of the jointdensity is not a rectangle with sides parallel to the axes. Then the domains ofintegration depend on the values of the conditioning variable.Example 3.4.4 (A Density with Weird Support).Consider the function{x+y 2 , x > 0,y>0,x+y0and y>0 and x + y
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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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Page 89 and 90: 2.7. INDEPENDENCE 81Show that the f
Page 91 and 92: Chapter 3Conditional Probability an
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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 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
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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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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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