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1、ProbabilityandRandomProcesses forElectricalEngineers JohnA?Gubner UniversityofWisconsin?Madison DiscreteRandomVariables Bernoulli?p? ? ?X?p? ? ?X?p? E?X?p?var?X?p?p?G X ?z?p?pz? binomial?n?p? ? ?X?k? ? n k ? p k ?p? n?k ?k?n? E?X?np?var?X?np?p?G X ?z?p?pz? n ? geometric ? ?p? ? ?X?k?p?p k ?k? E?X? p
2、 ?p ?var?X? p ?p? ? ?G X ?z? ?p ?pz ? geometric ? ?p? ? ?X?k?p?p k? ?k? E?X? ? ?p ?var?X? p ?p? ? ?G X ?z? ?p?z ?pz ? negativebinomialorPascal?m?p? ? ?X?k? ? k? m? ? ?p? m p k?m ?k?m?m? E?X? m ?p ?var?X? mp ?p? ? ?G X ?z? ? ?p?z ?pz ? m ? NotethatPascal?p?isthesameasgeometric ? ?p? Poisson? ? ?X?k?
3、? k e ? k? ?k? E?X?var?X?G X ?z?e ?z? ? FourierTransforms FourierTransform H?f? Z ? ? h?t?e ?j?ft dt InversionFormula h?t? Z ? ? H?f?e j?ft df h?t?H?f? I ?T?T? ?t?T sin?Tf? ?Tf ?W sin?Wt? ?Wt I ?W?W? ?f? ?jtj?T?I ?T?T? ?t?T ? sin?Tf? ?Tf ? ? W ? sin?Wt? ?Wt ? ? ?jfj?W?I ?W?W? ?f? e ?t u?t? ? ?j?f e
4、?jtj ? ? ? ?f? ? ? ? ? ?t ? ?e ?jfj e ?t? ? ? p ?e ? ? ?f? ? ? Preface IntendedAudience Thisbookcontainsenoughmaterialtoserveasatextforatwo?course sequenceinprobabilityandrandomprocessesforelectricalengineers?Itisalso usefulasareferencebypracticingengineers? Forstudentswithnobackgroundinprobabilitya
5、ndrandomprocesses?a ?rstcoursecanbeo?eredeitherattheundergraduateleveltotalentedjuniors andseniors?oratthegraduatelevel?Theprerequisiteistheusualundergrad? uateelectricalengineeringcourseonsignalsandsystems?e?g?HaykinandVan Veen?orOppenheimandWillsky?seetheBibliographyattheendof thebook? Asecondcour
6、secanbeo?eredatthegraduatelevel?Theadditionalprereq? uisiteissomefamiliaritywithlinearalgebra?e?g?matrix?vectormultiplication? determinants?andmatrixinverses?Becauseofthespecialattentionpaidto complex?valuedGaussianrandomvectorsandrelatedrandomvariables?the textwillbeofparticularinteresttostudentsin
7、wirelesscommunications?Ad? ditionally?thelastchapter?whichfocusesonself?similarprocesses?long?range dependence?andaggregation?willbeveryusefulforstudentsincommunication networkswhoareinterestedmodelingInternettra?c? MaterialforaFirstCourse Ina?rstcourse?Chapters?wouldmakeupthecoreofanyo?ering?These
8、chapterscoverthebasicsofprobabilityanddiscreteandcontinuousrandom variables?FollowingChapter?additionaltopicssuchaswide?sensestationary processes?Sections?thePoissonprocess?Section?discrete?time Markovchains?Section?andcon?denceintervals?Sections?can alsobeincluded?Thesetopicscanbecoveredindependent
9、lyofeachother?in anyorder?exceptthatProblem?inChapter?referstothePoissonprocess? MaterialforaSecondCourse Inasecondcourse?Chapters?and?wouldmakeupthecore?with additionalmaterialfromChapters?and?dependingonstudentprepa? rationandcourseobjectives?Forreviewpurposes?itmaybehelpfulatthebe? ginningoftheco
10、ursetoassignthemoreadvancedproblemsfromChapters? thataremarkedwitha ? ? Features Thosepartsofthebookmentionedaboveasbeingsuitablefora?rstcourse arewrittenatalevelappropriateforundergraduates?Moreadvancedproblems andsectionsinthesepartsofthebookareindicatedbya ? ?Thosepartsofthe booknotindicatedassui
11、tablefora?rstcoursearewrittenatalevelsuitable iii ivPreface forgraduatestudents?Throughoutthetext?therearenumericalsuperscripts thatrefertonotesattheendofeachchapter?Thesenotesareusuallyrather technicalandaddresssubtletiesofthetheory? Thelastsectionofeachchapterisdevotedtoproblems?Theproblemsare sep
12、aratedbysectionsothatallproblemsrelatingtoaparticularsectionare clearlyindicated?Thisenablesthestudenttorefertotheappropriatepart ofthetextforbackgroundrelatingtoparticularproblems?anditenablesthe instructortomakeupassignmentsmorequickly? TablesofdiscreterandomvariablesandofFouriertransformpairsaref
13、ound insidethefrontcover?Atableofcontinuousrandomvariablesisfoundinside thebackcover? Whencdfsorotherfunctionsareencounteredthatdonothaveaclosedform? Matlabcommandsaregivenforcomputingthem?Foralist?see?Matlab?in theindex? Theindexwascompiledasthebookwasbeingwritten?Hence?thereare manypointerstospeci
14、?cinformation?Forexample?see?noncentralchi?squared randomvariable? Withtheimportanceofwirelesscommunications?itisvitalforstudentsto becomfortablewithcomplexrandomvariables?especiallycomplexGaussian randomvariables?Hence?Section?isdevotedtothistopic?withspecial attentiontothecircularlysymmetriccomple
15、xGaussian?Alsoimportantinthis regardarethecentralandnoncentralchi?squaredrandomvariablesandtheir squareroots?theRayleighandRicerandomvariables?Theserandomvariables? aswellasrelatedonessuchasthebeta?F?Nakagami?andstudent?st?appear innumerousproblemsinChapters?and? Chapter?givesanextensivetreatmentofconvergenceinmeanoforder p?Specialattentionisgiventomean?squareconvergenceandtheHilbertspace ofsquare?integrablerandomvariables?Thisallowsustoprovetheprojection theorem?whichisimportantforestablishingtheexistenceofcertainestimators? andconditionalexpectationinpartic
