ele 509 introduction to random processes fall 2019swaszek/ele509/2019_01.pdf– general random...
TRANSCRIPT
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ELE509IntroductiontoRandomProcesses
Fall2019
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Iranthesameroutebothdays,what’sgoingon?
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Whatdothecirclesreallymean? Howaretheirsizesset?
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ModernApplications
• Mobility-on-demand(resourcemanagement)• Circuitdesign(failures)• Wirelesscommunications(channeleffects)• Medicaldiagnosis• Portfoliomanagement• Infrastructurecontrol• Onlinesearch• Voicerecognition• …
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AdministrativeIssues
• Classschedule• Officehours• Website• Classplan• Grading• Books• Neededskills:– Linearalgebra,calculus,programming
• UseofMatLab
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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– …
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Probability
• Intuitivenotion– cointoss– carddrawnfromdeck– lotterynumber
• Fractionvspercentage• “Even”chances• Complementprobability• Additivityifnooverlap– Subtractoffdoublecountedotherwise
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• Somesimpleexamples–Winningatroulette– Rollinga7with2dice– Generatinga“random”digitinMatLabusingrandi(k)
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• PlayerAthrows6dieandwinsifatleastone“1”showsup;PlayerBthrows12diceandwinsifatleasttwo“1”sshowup.Whohasabetterchanceofwinning?
• SupposedlyaskedofIsaacNewton
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• Aparkinglothas12spaces;8carsenterthelotandpark.Anobservernotesthatthe4emptyspacesareadjacenttoeachother.Isthisunexpected?
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• Winningatcraps– Rules:• Firstroll:winon7,11,loseon2,3,12,orcontinueon“point”
• Subsequentrolls:loseon7,winonpoint,continueotherwise
– Isitbettertorollorbethehouse?
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MatLab
• Generatingrandomnumbers:– randi,rand,randn– histogram
• Suggestions:– Small“samplesize”first
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• PlayerAthrows6dieandwinsifatleastone“1”showsup;PlayerBthrows12diceandwinsifatleasttwo“1”sshowup.Whohasabetterchanceofwinning?
• SupposedlyaskedofIsaacNewton
![Page 14: ELE 509 Introduction to Random Processes Fall 2019swaszek/ele509/2019_01.pdf– General random processes ... • “Even” chances • Complement probability • Additivity if no](https://reader033.vdocuments.net/reader033/viewer/2022042103/5e80fad2bbde9517e22200e4/html5/thumbnails/14.jpg)
• Aparkinglothas12spaces;8carsenterthelotandpark.Anobservernotesthatthe4emptyspacesareadjacenttoeachother.Isthisunexpected?
![Page 15: ELE 509 Introduction to Random Processes Fall 2019swaszek/ele509/2019_01.pdf– General random processes ... • “Even” chances • Complement probability • Additivity if no](https://reader033.vdocuments.net/reader033/viewer/2022042103/5e80fad2bbde9517e22200e4/html5/thumbnails/15.jpg)
• Winningatcraps– Rules:• Firstroll:winon7,11,loseon2,3,12,orcontinueon“point”
• Subsequentrolls:loseon7,winonpoint,continueotherwise
– Isitbettertorollorbethehouse?
![Page 16: ELE 509 Introduction to Random Processes Fall 2019swaszek/ele509/2019_01.pdf– General random processes ... • “Even” chances • Complement probability • Additivity if no](https://reader033.vdocuments.net/reader033/viewer/2022042103/5e80fad2bbde9517e22200e4/html5/thumbnails/16.jpg)
MontyHallProblem
• Gameshow(3choices;onegood,twobad)– Contestantselectsone– Shownoneoftheothers(abadprize)– Switchornot
• Approaches:– “oddshavechanged”– “nonewinformationsincethisisalwayspossible”
• Simulation– Generaterandomprizelocation– Generaterandomselection– Simulategame
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MatLabSimulation
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NeedaMathematicalApproachtoAnalysis
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SetTheoryReview
• Venndiagram• Universe,emptyset,subsets• Equality• Complements• Mutuallyexclusive,disjoint• Unionsandintersections
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• Unionsandintersections– Associative,commutative,distributivelaws(algebradefinestheseoperationsona“field”)
– DeMorgan’slaws– Powerset– Partitions
• Sigmaalgebra
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NextWeek
• Monday• chapter1oncombinatorialanalysis• problems-seewebsite
• Wednesday• chapter2• problems-seewebsite