ELE509IntroductiontoRandomProcesses

Fall2019

Iranthesameroutebothdays,what’sgoingon?

Whatdothecirclesreallymean? Howaretheirsizesset?

ModernApplications

• Mobility-on-demand(resourcemanagement)• Circuitdesign(failures)• Wirelesscommunications(channeleffects)• Medicaldiagnosis• Portfoliomanagement• Infrastructurecontrol• Onlinesearch• Voicerecognition• …

AdministrativeIssues

• Classschedule• Officehours• Website• Classplan• Grading• Books• Neededskills:– Linearalgebra,calculus,programming

• UseofMatLab

Topics• Today:concepts,useof

MatLab• StartingnextweekinRoss:

– Chap1:counting– Chap2:probability– Chap3:conditioning– Chap4:discreterandom

variables– Chap5:continuous

randomvariables– Chap6:randomvectors– Chap7:expectation– Chap8:limits

• Multiplesources(remainderofsemester):– Morerigorousmath

development– Generalrandomprocesses– Gaussianprocesses– Poissonprocesses– Markovprocesses/chains– …

Probability

• Intuitivenotion– cointoss– carddrawnfromdeck– lotterynumber

• Fractionvspercentage• “Even”chances• Complementprobability• Additivityifnooverlap– Subtractoffdoublecountedotherwise

• Somesimpleexamples–Winningatroulette– Rollinga7with2dice– Generatinga“random”digitinMatLabusingrandi(k)

• PlayerAthrows6dieandwinsifatleastone“1”showsup;PlayerBthrows12diceandwinsifatleasttwo“1”sshowup.Whohasabetterchanceofwinning?

• SupposedlyaskedofIsaacNewton

• Aparkinglothas12spaces;8carsenterthelotandpark.Anobservernotesthatthe4emptyspacesareadjacenttoeachother.Isthisunexpected?

• Winningatcraps– Rules:• Firstroll:winon7,11,loseon2,3,12,orcontinueon“point”

• Subsequentrolls:loseon7,winonpoint,continueotherwise

– Isitbettertorollorbethehouse?

MatLab

• Generatingrandomnumbers:– randi,rand,randn– histogram

• Suggestions:– Small“samplesize”first

• PlayerAthrows6dieandwinsifatleastone“1”showsup;PlayerBthrows12diceandwinsifatleasttwo“1”sshowup.Whohasabetterchanceofwinning?

• SupposedlyaskedofIsaacNewton

• Aparkinglothas12spaces;8carsenterthelotandpark.Anobservernotesthatthe4emptyspacesareadjacenttoeachother.Isthisunexpected?

• Winningatcraps– Rules:• Firstroll:winon7,11,loseon2,3,12,orcontinueon“point”

• Subsequentrolls:loseon7,winonpoint,continueotherwise

– Isitbettertorollorbethehouse?

MontyHallProblem

• Gameshow(3choices;onegood,twobad)– Contestantselectsone– Shownoneoftheothers(abadprize)– Switchornot

• Approaches:– “oddshavechanged”– “nonewinformationsincethisisalwayspossible”

• Simulation– Generaterandomprizelocation– Generaterandomselection– Simulategame

MatLabSimulation

NeedaMathematicalApproachtoAnalysis

SetTheoryReview

• Venndiagram• Universe,emptyset,subsets• Equality• Complements• Mutuallyexclusive,disjoint• Unionsandintersections

• Unionsandintersections– Associative,commutative,distributivelaws(algebradefinestheseoperationsona“field”)

– DeMorgan’slaws– Powerset– Partitions

• Sigmaalgebra

NextWeek

• Monday• chapter1oncombinatorialanalysis• problems-seewebsite

• Wednesday• chapter2• problems-seewebsite