# Activity 64 x y has the joint density function f x y

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Activity 6.4(),X Yhasthe jointdensityfunction( ,)f x ygivenby21211( ,)exp( ,)221f x yQ x y =where12,0andQisthefollowingquadraticform22112221221( ,)2(1)1xxyyQ x y=+.Showthat :(a)211(,)XN and222(,)YN,(b)thecorrelationbetweenXandYis,(c)XandYareindependentiff=0.
STAT301Probability Distributions94SummaryAfter completingthissection,youhavelearnttocomputeconditionalexpectation fromthebivariatenormaldensitytocalculatemomentsofthebivariatenormaldistribution.Assignment3 (a)Acontinuousbivariaterandomvariable(),X Yhasjointprobabilitydensityfunction(p.d.f)( ,)f x yforxandyreal.LetZXY=+. ShowthattheprobabilitydensityfunctionofZisgivenby( )( ,)k zf x zx dx−=Hencefind( )k zif2, 01 ,02/ 3( ,),elsewhere;0xyxxyf x y+=andcompute01PZ.Also, ifVXY=, findthejointp.d.fof(,)Z Vandhencethemarginalp.d.fofV.Assignment 3(b)andareindependentrandomvariableswithacommondensityfunctiongivenbyLetand.Findthejointp.d.fofand.Hencefindthemarginalp.d.f’sofand.UnitSummaryAlltoo soon,wehavecome totheendofthisUnit.Letmerecap whatyouhavelearntinthissection.Youhave learnthowto derivethedistributionoffunctionsofdiscrete randomvariableshow to derivethedistributionoffunctionsofcontinuous randomvariablesto calculateofmeanandvarianceoffunctionsofdiscrete randomvariables1X1X1 , 01( )0 , otherwise.xf x=112YXX=+212YXX=1Y2Y1Y2Y
STAT301Probability Distributions96Unit4GeneratingFunctionsandtheirApplicationsIntroductionInformally, thereare fourmain generatingfunctionsnamely;(i)The moment generating functions(ii)The factorial moment generating function(iii)Probability generating functions(iv)Characteristics functionswhere.ThisUnitwillcoverthefollowingtopics:Section1FactorialmomentgeneratingfunctionsSection2ProbabilitygeneratingfunctionsSection3MomentgeneratingfunctionsSection4ThecharacteristicfunctionsSection5TheReproductivePropertySection 6TheJointprobabilitygeneratingfunctionObjectivesAttheendofthisUnit ,youshouldbeabletocomputethefour generatingfunctions;tousetheuniquenessproperty to obtaindistributionofrandomvariablestousethegeneratingfunctionstoestimatemomentsofrandomvariables.()( )tXXE eMt=()( )itXE et=1i= −
STAT301Probability Distributions97Section1FactorialMomentGeneratingFunctionsIntroductionWelcometoSection1ofUnit4, generatingfunctions.Iwill discussinthisSectionageneratingfunctionforonly integer. Youwilllearntthatitisverygoodforgeneratingmoments.

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