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Probability and Stochastic Processes - Bucknell …

ProbabilityandStochasticProcessesA FriendlyIntroductionforElectricalandComp uterEngineersChapter2 Viewgraphs1 RandomVariablesExperiment:Procedure+ ObservationsObservationis anoutcomeAssigna numbertoeachoutcome:Randomvariable2 RandomVariablesThreewaystogeta rv:Thervis theobservationThervis a functionoftheobservationThervis a functionofa rv3 DiscreteRandomVariables rangeof(setofpossiblevalues)is discreteis is countableDiscretervhasPMF 4 PMFP roperties Foranevent , 5 BernoulliRVGetthephonenumberofa ,let. otherwise6 BinomialRVTestcircuits,eachcircuitis thenumberofsuccessesintrials: thenumberoftestsuptoandincludingthefirst reject. Fromthetree,,, otherwise8 Geometric: (y) ,, otherwise11 Pascal:, (l) rate, interval , otherwise14 Poisson: (j)15 Poisson: (j)16 Cumulative DistributionFunctionsThecumulativedistri butionfunction(CDF)ofrandomvariableis 17 CDFE xample (r) (r)Atthediscontinuitiesand, is theuppervalues.(righthandlimit)18 CDFP ropertiesForany discreterv, range satisfying , and Forall, For andsmall, for 19 ExpectedValueTheexpectedvalueofis : Each.

Probability and Stochastic Processes A Friendly Introduction for Electrical and Computer Engineers Chapter 2 Viewgraphs 1

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Transcription of Probability and Stochastic Processes - Bucknell …

1 ProbabilityandStochasticProcessesA FriendlyIntroductionforElectricalandComp uterEngineersChapter2 Viewgraphs1 RandomVariablesExperiment:Procedure+ ObservationsObservationis anoutcomeAssigna numbertoeachoutcome:Randomvariable2 RandomVariablesThreewaystogeta rv:Thervis theobservationThervis a functionoftheobservationThervis a functionofa rv3 DiscreteRandomVariables rangeof(setofpossiblevalues)is discreteis is countableDiscretervhasPMF 4 PMFP roperties Foranevent , 5 BernoulliRVGetthephonenumberofa ,let. otherwise6 BinomialRVTestcircuits,eachcircuitis thenumberofsuccessesintrials: thenumberoftestsuptoandincludingthefirst reject. Fromthetree,,, otherwise8 Geometric: (y) ,, otherwise11 Pascal:, (l) rate, interval , otherwise14 Poisson: (j)15 Poisson: (j)16 Cumulative DistributionFunctionsThecumulativedistri butionfunction(CDF)ofrandomvariableis 17 CDFE xample (r) (r)Atthediscontinuitiesand, is theuppervalues.(righthandlimit)18 CDFP ropertiesForany discreterv, range satisfying , and Forall, For andsmall, for 19 ExpectedValueTheexpectedvalueofis : Each.

2 If each occurs times, 21 DerivedRandomVariablesEachsamplevalueofa derivedrvis a functionofa samplevalueofa ,observe , ! rec d packet is error-free,rec vrsendsbackACK, ,thepacket is transmissionis , packet is sentEachpacket takes1 d untilthepacket is "?24 ExpectedvalueofThm:GivenrvwithPMF , theexpectedvalueof, is Example:: 25 VarianceandStdDeviationVariance: VariancemeasuresspreadofPMFS tandardDeviation: Unitsof ,.If, .27 ConditionalPMFofgivenGiven, with, Two version1 Probabilitymodeltellsus #forpossible .Example:Inthethmonthoftheyear, thenumberofcarscrossingtheGWbridgeis Poissonwithparam .29 ConditionalPMFs- version2is a subsetof suchthatforeach , eitheror. $ $% &otherwise30 ConditionalPMFE xampleExample:is geometricwith. Whatis theconditionalPMFofgiveneventthat?31 ConditionalExpectationsReplace with 32


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