Stat 5101 Lecture Notes - School of Statistics

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20 Stat 5101 (Geyer) Course Notes• cannot be a function of the dummy variable of integration x, and• is a function of the free variable y.Thus∫f X,Y (x, y) dx = some function of y onlyand hence can only be f Y (y) and cannot be f X (x). Thus making the mistakeof integrating with respect to the wrong variable (or variables) in attemptingto produce a marginal is really dumb on two counts: first, you were warnedbut didn’t get it, and, second, it’s not only a mistake in probability theory butalso a calculus mistake. I do know there are other reasons people can makethis mistake, being rushed, failure to read the question, or whatever. I knowsomeone will make this mistake, and I apologize in advance for insulting youby calling this a “dumb mistake” if that someone turns out to be you. I’m onlytrying to give this lecture now, when it may do some good, rather than later,written in red ink all over someone’s test paper. (I will, of course, be shockedbut very happy if no one makes the mistake on the tests.)Of course, we sum out discrete variables and integrate out continuous ones.So how do we go from f W,X,Y,Z to f X,Z ? We integrate out the variables wedon’t want. We are getting rid of W and Y ,so∫∫f X,Z (x, z) = f W,X,Y,Z (w, x, y, z) dw dy.If the variables are discrete, the integrals are replaced by sumsf X,Z (x, z) = ∑ ∑f W,X,Y,Z (w, x, y, z).w yIn principle, it couldn’t be easier. In practice, it may be easy or tricky, dependingon how tricky the problem is. Generally, it is easy if there are no worries aboutdomains of integration (and tricky if there are such worries).Example 1.5.1.Consider the distribution of Example 1.3.1 with joint density of X and Y givenby (1.22). What is the marginal distribution of Y ? We find it by integratingout X∫∫ 1∣f Y (y) = f(x, y) dx = (x + y) dx = x2 ∣∣∣1 ( ) 12 + xy =2 + y0Couldn’t be simpler, so long as you don’t get confused about which variableyou integrate out.That having been said, it is with some misgivings that I even mention thefollowing examples. If you are having trouble with joint and marginal distributions,don’t look at them yet! They are tricky examples that very rarely arise.If you never understand the following examples, you haven’t missed much. Ifyou never understand the preceding example, you are in big trouble.0
1.5. JOINT AND MARGINAL DISTRIBUTIONS 21Example 1.5.2 (Uniform Distribution on a Triangle).Consider the uniform distribution on the triangle with corners (0, 0), (1, 0), and(0, 1) with densityf(x, y) =2, 0
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: 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 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
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Page 61 and 62: 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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2.5. 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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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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