Stat 5101 Lecture Notes - School of Statistics

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182 Stat 5101 (Geyer) Course NotesThat is, when we put parentheses around the subscripts, that means we have putthe values in ascending order. For any real number x, the notation ⌈x⌉ denotesthe smallest integer greater than or equal to x, which is called the ceiling of x,and the notation ⌊x⌋ denotes the largest integer less than or equal to x, whichis called the floor of x,Theorem 7.5. If np is not an integer, then the p-th quantile of the empiricaldistribution associated with the vector x is unique and is equal to x (⌈np⌉) .When np is an integer, then any point x such thatis a p-th quantile.x (np) ≤ x ≤ x (np+1) (7.9)Proof. The p-th quantile must be a point x such that there are at least np ofthe x i at or below x and at least n(1 − p) atorabovex.In the case that np is not an integer, let k = ⌈np⌉. Since np is not an integer,and ⌈np⌉ is the least integer greater than k, we have k>np>k−1. What wemust show is that x (k) is the unique p-th quantile.There are at least k>npdata pointsx (1) ≤···≤x (k)at or below x (k) . Furthermore, if ik, then n − i +1≤n−k
7.1. EMPIRICAL DISTRIBUTIONS 183To find the 0.25 quantile, also called the 25-th percentile, the theorem says wefind ⌈np⌉, which is the integer above np =2.5, which is 3, and then the empiricalquantile is the corresponding order statistic, that is x (3) =0.29.We remark in passing that if the 25-th percentile is 3 in from the left end ofthe data in sorted order, then the 75-th percentile is 3 in from the right end, sothe definition behaves as we expect. Let’s check this. First np =7.5. Roundingup gives 8. And x (8) =1.02 is indeed the third from the right.The definition gets tricky is when np is an integer. If we want the 40-thpercentile, np = 4. Then the theorem says that any point x between x (4) =0.35and x (5) =0.49 is a 40-th percentile (0.4 quantile) of these data. For example,0.35, 0.39, 0.43, and 0.49 are all 40-th percentiles. A bit weird, but that’s howthe definition works.7.1.6 The Empirical MedianThe median of the empirical distribution we denote by ˜x n . It is the p-thquantile for p =1/2. By the theorem, the median is unique when np is notan integer, which happens whenever n is an odd number. When n is an evennumber, the empirical median is not unique. It is any point x satisfying (7.9),where k = n/2. This nonuniqueness is unsettling to ordinary users of statistics,so a convention has grown up of taking the empirical median to be the midpointof the interval given by (7.9).Definition 7.1.4 (Empirical Median).The median of the values x 1 , ..., x n is the middle value in sorted order whenn is odd˜x n = x (⌈n/2⌉)and the average of the two middle values when n is even˜x n = x (n/2) + x (n/2+1)2Example 7.1.6.The data in Example 7.1.1 have 10 data points. So we are in the “n even” case,and the empirical median is the average of the two middle values of the data insorted order, that is,˜x n = x (5) + x (6)2=0.49 + 0.7727.1.7 Characterization of the Median=0.63Corollary 7.6 (Characterization of the Median). If X is a real-valuedrandom variable having finite expectation, then a median of X is any value ofa that minimizes the functiong(a) =E{|X − a|}which is the expected absolute deviation from 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
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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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2.4. MOMENTS 49What happens to Coro
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2.4. MOMENTS 51This inequality is a
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2.4. MOMENTS 53(a)Why does (2.7) as
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2.5. PROBABILITY THEORY AS LINEAR A
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2.6. PROBABILITY IS A SPECIAL CASE
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2.7. INDEPENDENCE 772.7 Independenc
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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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Page 183 and 184: 6.1. UNIVARIATE THEORY 175Problems6
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Page 219 and 220: Appendix AGreek LettersTable A.1: T
Page 221 and 222: Appendix BSummary of Brand-NameDist
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Page 225 and 226: B.2. CONTINUOUS DISTRIBUTIONS 217Th
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Page 237 and 238: Appendix DRelations Among BrandName
Page 239 and 240: 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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