The Theory of Probability: Explorations and Applications (Instructor Solution Manual, Selected Solutions)

434 Pages·2015·2.229 MB·English

by  Santosh S. Venkatesh

Problems for Solution
from
The Theory of Probability
and
Selected Solutions
SantoshS.Venkatesh
UniversityofPennsylvania

Contents
Preface vii
A ELEMENTS
I ProbabilitySpaces 3
II ConditionalProbability 25
III AFirstLookatIndependence 51
IV ProbabilitySieves 65
V NumbersPlayaGameofChance 87
VI TheNormalLaw 105
VII ProbabilitiesontheRealLine 121
VIII TheBernoulliSchema 143
IX TheEssenceofRandomness 169
X TheCodaoftheNormal 211
B FOUNDATIONS
XI DistributionFunctionsandMeasure 247
XII RandomVariables 257
XIII GreatExpectations 279
XIV VariationsonaThemeofIntegration 293
Contents
XV LaplaceTransforms 331
XVI TheLawofLargeNumbers 357
XVII FromInequalitiestoConcentration 377
XVIII PoissonApproximation 385
XIX ConvergenceinLaw,SelectionTheorems 397
XX NormalApproximation 407
Index 419
vi
Preface
Several generations of students and teaching assistants worked out problems
and contributed to the selected solutions assembled here. I should especially
acknowledgeWeiBi,ShaoChiehFang,GauravKasbekar,JonathanNukpezah,
AlirezaTahbazSalehi,ShahinShahrampour,EvangelosVergetis,andZhengwei
Wu. I must confess immediately that not all the solutions have seen the same
level of scrutiny; and, while I have expended some effort to bring some uni-
formityofstyleandpresentationtothematerial,itmayoccasionallylacksome
polish. Itstruckme,however,thatareaderdesirousofanearlyviewofthere-
sultsmaybewillingtoforgivethedeficienciesinthecurrentcrudeassemblage
and,accordingly,thatitmaybepreferabletoprovidearoughandreadycollec-
tionofsolutionsnow,howeverunpolished,unvetted,andincomplete,andnot
waittillIhadhadthetimetoproof-readthemcarefullyandpresentthemina
uniformformatandstylewhilefillinginthegaps.
ReferencestoToPare,ofcourse,toTheTheoryofProbability; references
to Equations, Examples, Theorems, Sections, Chapters, and so on point to the
maintext;thereferencingconventionsareasinToP.Asanavigationalaid,page
headings are labelled by chapter and problem; the detailed index that I have
providedmayalsobeofuse.
Thereareinevitablyerrorsinthetextlyinginwaittobediscovered. I
can only hope that these are of the obvious kind that do not cause great con-
sternation and apologise for these in advance. I would certainly very much
appreciatereceivingwordoferrorsandambiguitiesbothhereandinthemain
textofToP.
January 1, 2015: The task of compiling solutions proceeds slowly. The
current version of the palimpsest provides solutions for more than 75% of the
problems: ofthe560problemsassignedforsolutioninToP,428nowhavesolu-
tionsspelledoutindetail. InmanyinstancesIhaveprovidedtwoorthreedif-
ferent approaches to solution to illustrate different perspectives; and the solu-
tionsfrequentlyfleshoutthemesandgeneralisationssuggestedbythepatterns
of attack. Table 1 on the following page identifies the current list of problems
whosesolutionsappearinthismanuscript.
Preface
Chaptertitles Solutionsprovided Notreadyforprimetime
A I ProbabilitySpaces 1–30 31–34
II ConditionalProbabilities 1–19,22–31 20,21
III AFirstLookatIndependence 1–20 21–23
IV ProbabilitySieves 1–26,29 27,28
V NumbersPlayaGameofChance 1–18 19–21
VI TheNormalLaw 1–11 12,13
VII ProbabilitiesontheRealLine 1–22 —
VIII TheBernoulliSchema 1–26,28–38 27
IX TheEssenceofRandomness 1–36 —
X TheCodaoftheNormal 1–12,14–22,24–33 13,23
B XI DistributionFunctionsandMeasure 1–5,7–11 6,12,13
XII RandomVariables 1–24 25,26
XIII GreatExpectations 1–5,7–23 6,24–26
XIV VariationsonaThemeofIntegration 1–11,13–22,24–27,30–36,39–51,54–56 12,23,28,29,37,38,52,53
XV LaplaceTransforms 1–11,17,20–31 12–16,18,19
XVI TheLawofLargeNumbers 1–4,11,14–20 5–10,12,13,21–31
XVII FromInequalitiestoConcentration 1 2–35
XVIII PoissonApproximation 4,5,20–23 1–3,6–19
XIX ConvergenceinLaw,SelectionTheorems 1–4,6,11,17,19,20 5,7–10,12–16,18
XX NormalApproximation 1–3,6–8,10–14 4,5,9,15–19
C XXI Sequences,Functions,Spaces — —
Table1:Listofcompiledsolutions.
viii
Part A
E
LEMENTS