Stat 5101 Lecture Notes - School of Statistics

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40 Stat 5101 (Geyer) Course Notes2.4.2 Second Moments and VariancesThe preceding section says all that can be said in general about first moments.As we shall now see, second moments are much more complicated.The most important second moment is the second central moment, whichalso has a special name. It is called the variance and is often denoted σ 2 . (Thesymbol σ is a Greek letter. See Appendix A). We will see the reason for thesquare presently. We also use the notation var(X) for the variance of X. Soσ 2 =µ 2 = var(X) =E{(X−µ) 2 }.As we did with means, when there are several random variables under discussion,we denote the variance of each using the same Greek letter σ, but add thevariable as a subscript to distinguish them: σX 2 = var(X), σ2 Y = var(Y ), and soforth.Note that variance is just an expectation like any other, the expectation ofthe random variable (X − µ) 2 .All second moments are related.Theorem 2.11 (Parallel Axis Theorem). If X is a random variable withmean µ and variance σ 2 , thenE{(X − a) 2 } = σ 2 +(µ−a) 2Proof. Using the factfrom algebra(b + c) 2 = b 2 +2bc + c 2 (2.14)(X − a) 2 =(X−µ+µ−a) 2=(X−µ) 2 +2(X−µ)(µ − a)+(µ−a) 2Taking expectations of both sides and applying linearity of expectation (everythingnot containing X is nonrandom and so can be pulled out of expectations)givesE{(X − a) 2 } = E{(X − µ) 2 } +2(µ−a)E(X−µ)+(µ−a) 2 E(1)= σ 2 +2(µ−a)µ 1 +(µ−a) 2By Theorem 2.9, the middle term on the right hand side is zero, and thatcompletes the proof.The name of this theorem is rather strange. It is taken from an analogoustheorem in physics about moments of inertia. So the name has nothing to dowith probability in general and moments (as understood in probability theoryrather than physics) in particular, and the theorem is not commonly calledby that name. We will use it because Lindgren does, and perhaps becausethe theorem doesn’t have any other widely used name. In fact, since it is so
2.4. MOMENTS 41simple, it is often not called a theorem but just a calculation formula or method.Sometimes it is called “completing the square” after the method of that namefrom high-school algebra, although that name isn’t very appropriate either. Itis a very simple theorem, just the algebraic identity (2.14), which is related to“completing the square” plus linearity of expectation, which isn’t. Whatever itis called, the theorem is exceedingly important, and many important facts arederived from it. I sometimes call it “the most important formula in statistics.”Corollary 2.12. If X is a random variable having first and second moments,thenvar(X) =E(X 2 )−E(X) 2 .The proof is left as an exercise (Problem 2-13).This corollary is an important special case of the parallel axis theorem. Italso is frequently used, but not quite as frequently as students want to use it.It should not be used in every problem that involves a variance (maybe in halfof them, but not all). We will give a more specific warning against overusingthis corollary later.There are various ways of restating the corollary in symbols, for exampleandσ 2 X = E(X 2 ) − µ 2 X,µ 2 = α 2 − α 2 1.As always, mathematics is invariant under changes of notation. The importantthing is the concepts symbolized rather than the symbols themselves.The next theorem extends Theorem 2.2 from means to variances.Theorem 2.13. Suppose X is a random variable having first and second momentsand a and b are real numbers, thenvar(a + bX) =b 2 var(X). (2.15)Note that the right hand side of (2.15) does not involve the constant part aof the linear transformation a + bX. Also note that the b comes out squared.The proof is left as an exercise (Problem 2-15).Before leaving this section, we want to emphasize an obvious property ofvariances.Sanity Check: Variances are nonnegative.This holds by the positivity axiom (E3) because the variance of X is the expectationof the random variable (X − µ) 2 , which is nonnegative because squaresare nonnegative. We could state this as a theorem, but won’t because its mainuse is as a “sanity check.” If you are calculating a variance and don’t makeany mistakes, then your result must be nonnegative. The only way to get anegative variance is to mess up somewhere. If you are using Corollary 2.12, for
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
Page 39 and 40: Chapter 2Expectation2.1 Introductio
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: 2.4. MOMENTS 39holds for all real-v
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 and 94: 3.1. PARAMETRIC FAMILIES OF DISTRIB
Page 95 and 96: 3.2. CONDITIONAL PROBABILITY DISTRI
Page 97 and 98: 3.3. AXIOMS FOR CONDITIONAL EXPECTA
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3.3. AXIOMS FOR CONDITIONAL EXPECTA
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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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