Topic 7: Random Processes

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Topic7: RandomProcesses De nition,discreteandcontinuousprocesses Specifyingrandomprocesses{Joint cdf's or pdf's{Mean,auto-covariance,auto-correlat ion{Cross-covariance,cross-correlation StationaryprocessesandergodicityES150{ Harvard SEAS1Randomprocesses Arandomprocess, alsocalledastochasticprocess, is a familyof randomvariables,indexedby a parametertfromanindexingsetT. For eachexperiment outcome!2 ,we assigna functionXthatdependsontX(t; !)t2T; !2 {tis typicallytime,butcanalsobe a spatialdimension{tcanbe discreteor continuous{Therangeoftcanbe nite,butmoreoftenis in nite,which meanstheprocesscontainsanin nitenumber of randomvariables. Examples:{Thewirelesssignalreceivedby a cellphoneover time{Thedailystock price{Thenumber of packetsarrivingat a routerin 1-secondintervals{Theimageintensity over 1cm2regionsES150{ Harvard SEAS2 We areinterestedin specifyingthejoint behaviorof therandomvariableswithina family, or thebehaviorof a studying{Thedependenciesamongtherandomva riablesof theprocess( ){Long-termaverages{Extremeor boundaryevents ( ){Estimation/detectionof a signalcorruptedby noiseES150{ Harvard SEAS3Two ways of viewinga randomprocessConsidera processX(t; !)}}}}}}}}}}}}}}}}}

(e.g. for prediction) { Long-term averages { Extreme or boundary events (e.g. outage) { Estimation/detection of a signal corrupted by noise ES150 { Harvard SEAS 3 Two ways of viewing a random process Consider a process X(t;!) † At a flxed t, X(t;!) is …

  Processes, Topics, Prediction, Random, Topic 7, Random processes