Stat 5101 Lecture Notes - School of Statistics

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30 Stat 5101 (Geyer) Course Notesbecause f X happens to be a symmetric about zero, that is, f X (x) =f X (−x).Note that it is just as well we deleted the point zero at the beginning, becausethe resulting density is undefined at zero anyway.It is worthwhile stating a couple of intermediate results of the precedingexample in a corollary.Corollary 1.9. Suppose X is a continuous random scalar with density f X , thenY = X 2 has densityf Y (y) = 12 √ [fX ( √ y)+f X (− √ y) ] , y > 0.yMoreover, if f X is symmetric about zero, thenf Y (y) = 1 √ yf X ( √ y), y > 0.
Chapter 2Expectation2.1 IntroductionExpectation and probability are equally important concepts. An importanteducational objective of this course is that students become “ambidextrous” inreasoning with these two concepts, able to reason equally well with either.Thus we don’t want to think of expectation as a derived concept—somethingthat is calculated from probabilities. We want the expectation concept to standon its own. Thus it should have the same sort of treatment we gave probability.In particular, we need to have the connection between expectation and thelaw of large numbers (the analog of Section 2.2 in Lindgren) and axioms forexpectation (the analog of Section 2.4 in Lindgren).Suppose you are asked to pick a single number to stand in for a random variable.Of course, the random variable, when eventually observed, will probablydiffer from whatever number you pick (if the random variable is continuous itwill match whatever number you pick with probability zero). But you still haveto pick a number. Which number is best?The expectation (also called expected value) of a real-valued random variable,if it exists, is one answer to this problem. It is the single number that a rationalperson “should” expect as the value of the random variable when it is observed.Expectation is most easily understood in economic contexts. If the randomvariable in question is the value of an investment or other uncertain quantity,the expectation is the “fair price” of the investment, the maximum amount arational person is willing to pay to pay for the investment.The notion of expectation of a non-monetary random variable is less clear,but can be forced into the monetary context by an imaginary device. Supposethe random variable in question is the weight of a student drawn at randomfrom a list of all students at the university. Imagine you will be paid a dollarper pound of that student’s weight. How much would you be willing to pay to“invest” in this opportunity? That amount is (or should be) the expected valueof the student’s weight.31
Page 1 and 2: Stat 5101 Lecture NotesCharles J. G
Page 3 and 4: Contents1 Random Variables and Chan
Page 5 and 6: CONTENTSv5.1.4 Variance Matrices ..
Page 7 and 8: CONTENTSviiD Relations Among Brand
Page 9 and 10: Chapter 1Random Variables andChange
Page 11 and 12: 1.1. RANDOM VARIABLES 3Sometimes it
Page 13 and 14: 1.1. RANDOM VARIABLES 5that is, X i
Page 15 and 16: 1.2. CHANGE OF VARIABLES 7the union
Page 17 and 18: 1.2. CHANGE OF VARIABLES 9Thus even
Page 19 and 20: 1.2. CHANGE OF VARIABLES 11Every in
Page 21 and 22: 1.2. CHANGE OF VARIABLES 13where f
Page 23 and 24: 1.3. RANDOM VECTORS 151.3.1 Discret
Page 25 and 26: 1.4. THE SUPPORT OF A RANDOM VARIAB
Page 27 and 28: 1.5. JOINT AND MARGINAL DISTRIBUTIO
Page 29 and 30: 1.5. JOINT AND MARGINAL DISTRIBUTIO
Page 31 and 32: 1.6. MULTIVARIABLE CHANGE OF VARIAB
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Page 41 and 42: 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
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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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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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