Stat 5101 Lecture Notes - School of Statistics

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86 Stat 5101 (Geyer) Course Notes3.2 Conditional Probability DistributionsScalar VariablesThe conditional probability distribution of one random variable Y givenanother X is the probability model you are supposed to use in the situationwhen you have seen X and know its value but have not yet seen Y and don’tknow its value. The point is that X is no longer random. Once you know itsvalue x, it’s a constant not a random variable.We write the density of this probability model, the conditional distributionof Y given X as f(y | x). We write expectations with respect to this model asE(Y | x), and we write probabilities asP (Y ∈ A | x) =E{I A (Y)|x}(couldn’t resist an opportunity to reiterate the lesson of Section 2.6 that probabilityis a special case of expectation).We calculate probabilities or expectations from the density in the usual waywith integrals in the continuous case∫E{g(Y ) | x} = g(y)f(y | x) dy (3.3)∫P {Y ∈ A | x} = f(y | x) dy (3.4)and with the integrals replaced by sums in the discrete case.Note thatA conditional probability density is just an ordinary probability density,when considered as a function of the variable(s) in front of thebar alone with the variable(s) behind the bar considered fixed.This means that in calculating a conditional probability or expectation from aconditional densityalways integrate with respect to the variable(s) in front of the bar(with, of course, “integrate” replaced by “sum” in the discrete case).Example 3.2.1 (Exponential Distribution).Of course, one doesn’t always have to do an integral or sum, expecially when a“brand name” distribution is involved. Suppose the conditional distribution ofY given X is Exp(X), denotedAY | X ∼ Exp(X)for short. This means, of course, that the conditional density isf(y | x) =xe −xy , y > 0
3.2. CONDITIONAL PROBABILITY DISTRIBUTIONS 87(just plug in x for λ in the formula in Section B.2.2 in Appendix B), but wedon’t need to use the density to calculate the conditional expectation, becausewe know that the mean of the Exp(λ) distribution is 1/λ, hence (again justplugging in x for λorE(Y | x) = 1 xE(Y | X) = 1 Xdepending on whether we are thinking of the variable behind the bar as random(big X) or fixed (little x) As we shall see, both viewpoints are useful and weshall use both in different situations.If the known formulas for a “brand name” distribution don’t answer thequestion, then we do need an integralP (a
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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
Page 43 and 44: 2.3. BASIC PROPERTIES 35that is, th
Page 45 and 46: 2.4. MOMENTS 37The Multiplicativity
Page 47 and 48: 2.4. MOMENTS 39holds for all real-v
Page 49 and 50: 2.4. MOMENTS 41simple, it is often
Page 51 and 52: 2.4. MOMENTS 43In contrast, for all
Page 53 and 54: 2.4. MOMENTS 45is the sum of the ex
Page 55 and 56: 2.4. MOMENTS 47Proof. Just take a i
Page 57 and 58: 2.4. MOMENTS 49What happens to Coro
Page 59 and 60: 2.4. MOMENTS 51This inequality is a
Page 61 and 62: 2.4. MOMENTS 53(a)Why does (2.7) as
Page 63 and 64: 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
Page 91 and 92: Chapter 3Conditional Probability an
Page 93: 3.1. PARAMETRIC FAMILIES OF DISTRIB
Page 97 and 98: 3.3. AXIOMS FOR CONDITIONAL EXPECTA
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 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.
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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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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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