Stat 5101 Lecture Notes - School of Statistics

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96 Stat 5101 (Geyer) Course NotesOf course, there is a slight problem with (3.14) when the denominator is zero,but since the set of x such that f X (x) = 0 is a set of probability zero, this doesnot matter, and f(y | x) can be defined arbitrarily for all such x.Example 3.4.1 (Uniform Distribution on a Triangle).This continues Example 1.5.2. Recall from that example that if X and Y havejoint densitythat the marginal of X isf(x, y) =2, 0
3.4. JOINT, CONDITIONAL, AND MARGINAL 97only define conditional densities where the values are not arbitrary and actuallymatter. The only reason we mention this issue at all is so that you won’t thinkf(y | x) has to have a sensible definition for all possible x.So how about conditional expectations? Given the formula (3.15) for theconditional density, we just plug and chug∫E(Y | x) = yf(y | x) dy = 1 − x(3.16a)2∫E(Y 2 | x) = y 2 (1 − x)2f(y|x)dy = (3.16b)3var(Y | x) =E(Y 2 |x)−E(Y |x) 2 (1 − x)2= (3.16c)12and so forth, (3.16c) holding because of Corollary 2.12, which like every otherfact about unconditional expectation, also holds for conditional expectation solong as we are considering the conditioning variables fixed.We could end Section 3.4 right here. Formulas (3.13) and (3.14) tell us howto calculate conditionals from joints and joints from conditionals and marginals.And the fact that “conditional expectation is a special case of ordinary expectation”(so long as we are considering the conditioning variables fixed) tells howto compute expectations. So what else is there to know? Well, nothing, but alot more can be said on the subject. The rest of Section 3.4 should give you amuch better feel for the subject and allow you to calculate conditional densitiesand expectations more easily.3.4.2 NormalizationA standard homework problem for courses like this specifies some nonnegativefunction h(x) and then asks for what real number k is f(x) =kh(x) aprobability density.Clearly we must have k>0, because 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
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
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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 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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5.2. THE MULTIVARIATE NORMAL DISTRI
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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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