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ACCT620Chapter2

Course: ACCT 202, Spring 2011
School: NYU
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Thus,allocationof CHAPTER2 MEASURINGPRODUCTCOSTS Questions,Exercises,Problems,andCases:AnswersandSolutions 2.1 Seetextorglossaryattheendofthebook. 2.2 Underajobcostingsystem,costsareaccumulatedbyjob. these"costs"tooutputisrelativelysimplesincetheproductisawelldefined, specific customer order. Under process costing, costs are accumulated by...

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Thus,allocationof CHAPTER2 MEASURINGPRODUCTCOSTS Questions,Exercises,Problems,andCases:AnswersandSolutions 2.1 Seetextorglossaryattheendofthebook. 2.2 Underajobcostingsystem,costsareaccumulatedbyjob. these"costs"tooutputisrelativelysimplesincetheproductisawelldefined, specific customer order. Under process costing, costs are accumulated by departmentorproductionprocesses.Costsarethenspreadevenlyovertheunits produced. 2.3 Service organizations do not have a tangible "good." Therefore, there is no tangible item which would qualify as an inventory item. All of the costs of servicepersonnelareconsideredexpiredasincurred.Theorganization'sproduct serviceisprovidedintheperiodinwhichservicelaborcostsareincurred. 2.4 Aproductionoperationisastandardizedmethodofmakingaproduct thatis repeatedlyperformed. 2.5 Operationcostinghascharacteristicsofbothjobcostingandprocesscosting,so itiscalledahybridofthesetwo. 2.6 BeginningBalance+TransfersIn=TransfersOut+EndingBalance. 2.7 Assigningcoststothewrongjobsgivesmisinformationaboutthecostsofjobs. Thismisinformationaffectstheevaluationoftheperformanceofjobsupervisors. Itaffectsjobpricingifthejobispartiallyortotallycostpluspricing.Managers use cost information about past jobs to estimate the costs, and therefore the prices,offuturejobs.Misinformationaboutjobsaffectsthecostestimatesand pricesoffuturejobs. 2.8 Weagreewiththecontrollerinthissituation.Often,jobcostingistoodetailed andexpensivetooperateforroutinebatchesofhomogeneousgoods. 2.9 ForJITtobefeasible,acompanyshouldhavereliablesuppliersofproduction inputs, customers who are predictable in placing orders, quality production, workersskilledtoperformmultipletasks,andahighqualityworkethic. 21 Solutions 2.10 Justintimeallowscompaniestoreduceinventorylevelsandthetimebetween production and delivery. Lower inventory levels and reduced delivery time enablesaccountantstoexpensevirtuallyallcostsintheperiodinwhichtheyare incurred,whichreducesrecordkeeping,particularlyforinventories. Buusing JIT, companies have been known to save hundreds of thousands of journal entrieseveryyear. 2.11 Both service and manufacturing companies need good managerial accounting information; the difference in providing quality is in the timing. Service organizations do not produce inventory but deliver the service directlyto the customersothatdefectsarehardertoprevent. Manufacturingcompaniescan checkthequalityofproductsbeforetheyareshippedtocustomerssoerrorscan bedetectedandcorrected. 2.12 Acompanyusingoperationcostingwilltypicallyusedifferentmaterialsforeach typeofproduct,whichissimilartojobcostingwhereeachjoborbatchisunique. Thedifferentproductswillpassthroughoperationsinwhicheachproducthas thesameworkdoneonitintheoperation.Forexample,acompanymayinstall differentmaterialsasseatcoversinanautomobileleather,vinyl,orcloth.The operationofinstallingseatcoverscouldbeessentiallythesameforeachtypeof materialsotheapplicationoflaborandoverheadwouldbesimilartoprocess costing. Inpractice,differencesinmaterialscouldaffecttheoperation. Itis easy to imagine that particular materials would be harder to install, for example. The essential costing system would still be operation costing, nevertheless. 2.13 JITcansaveinventorycarryingcostsandaccountingrecordkeepingcosts. It alsomayreducecostsofproductionproblemssuchaspoorqualitythatcanbe hidden by keeping inventories and buffer stocks between production work stations. 2.14 UsingJIT, production costsareimmediatelyexpensed through Cost ofGoods Sold as those costs are incurred. If there are inventories at the end of a reportingperiod,theaccountantscreditCostofGoodsSoldanddebitinventory accountsto"backout"inventoryamountsfromCostofGoodsSold. 2.15 If a company maintains no inventories, it will have to shut down production wheneverasupplierdoesnotdeliverthepropermaterialsofthespecifiedquality attherighttime. 2.16 The manager of the Gravins Division reported overstated ending inventory levels to increase profits. However, after one period, he was faced with the dilemma of having to again overstate ending inventory so as to not reduce profits. Thissituationcontinueduntiltopmanagementnoticedtheunusually largeamountofendinginventoryanduncoveredthefraud. Solutions 22 2.17 Somecompaniesthatmakeproductsusingprocessesare: Husch(wine) BethlehemSteel(steel) Pillsbury(flourproducts) Kellogg(cereal) CommonGrounds(coffee) Heinz(catsup) MillerBrewingCompany(beer) 2.18 Somecompaniesthatproducejobsare: Accenture(consulting) GuyF.Atkinson(construction) Bechtel(engineering) Anyuniversity(researchgrants) ThomsonSouthWesternpublishingcompany(thisbook) Anyhospital(surgeries) UniversalStudios(movies) 2.19 (Costflowmodel.) Ingeneral,applythefollowingmodel: BB+TI = TO+EB Midwest: BB+$200,000 BB BB Northeast: $60,000+$200,000 = $220,000+EB EB = $60,000+$200,000$220,000 EB = $40,000 Southeast: BB+$160,000 = $150,000+$40,000 BB = $150,000+$40,000$160,000 BB = $30,000 = = = 23 $180,000+$60,000 $180,000+$60,000$200,000 $40,000 Solutions 2.20 (Costflowmodel.) Ingeneral,applythefollowingmodel: BB+TI = TO+EB Lighter Fluid: $40,000+$180,000 = EB = EB = $80,000+EB $40,000+$180,000$80,000 $140,000 Waterproof Matches: $60,000+$340,000 = EB = EB = $380,000+EB $60,000+$340,000$380,000 $20,000 Burn Ointment: $60,000+$120,000 = EB = EB = $140,000+EB $60,000+$120,000$140,000 $40,000 FireplaceScreens: Wecannotcomputebecausewehavetwounknownsinthebasiccostflow equation.Werequireeitherthebeginningorendinginventorytosolvethis partoftheproblem. 2.21 (Costflowmodel.) Ingeneral,applythefollowingmodel: BB+TI = TO+EB Rubber Rafts: $160,000+$180,000 = EB = EB = Rubber $60,000+$90,000 Duckies: EB EB = = = $240,000+EB $160,000+$180,000$240,000 $100,000 $110,000+EB $60,000+$90,000$110,000 $40,000 Galoshes: $60,000+$480,000 = $540,000+EB EB = $60,000+$480,000$540,000 EB = $0 DivingEquipment: Cannot compute because we have two unknowns in the basic cost flow equation.Werequireeitherthebeginningorendinginventorytosolvethis partoftheproblem. Solutions 24 2.22 (Costflowmodel.) Ingeneral,applythefollowingmodel: BB+TI = TO+EB tofindwhattheendinginventoryshouldbepertherecords. Video $20,000+$40,000 cassette EB Recorders: EB = = = $35,000+EB $20,000+$40,000$35,000 $25,000 $5,000(=$25,000$20,000physicalcount)worthofvideo cassetterecordersismissing. Televisions: $20,000+$50,000 EB EB = = = $55,000+EB $20,000+$50,000$55,000 $15,000 $10,000(=$15,000$5,000physicalcount)worthoftele visionsismissing. Compact Disc Players: $15,000+$20,000 EB EB = = = $25,000+EB $15,000+$20,000$25,000 $10,000 Nodiscrepancyincompactdiscplayers. 25 Solutions 2.23 (Costflowmodel.) Usethecostflowequation, BB+TI = TO+EB tofindwhattheendinginventoryshouldbepertherecords. Computer Chips: $600,000+$1,600,000=$1,800,000+EB EB =$600,000+$1,600,000$1,800,000 EB =$400,000 Thephysicalcountshows$200,000(=$600,000$400,000)more thanintherecords. Potato Chips: $160,000+$600,000 =$500,000+EB EB =$160,000+$600,000$560,000 EB =$260,000 $20,000(=$260,000$240,000physicalcount)discrepancy betweentherecordsandactualpotatochipsininventory. PokerChips $60,000+$200,000 =$180,000+EB EB =$60,000+$200,000$180,000 EB =$80,000 $30,000(=$80,000$50,000physicalcount)discrepancy betweentherecordsandactualpokerchipsininventory. Comment:Ingeneral,therearedifferencesbetweenwhatappearsonthebooks andwhatexistsininventorybecauseofinventoryshrinkageduetotheft, breakageorobsolescence.Also,therecouldbeerrorsinrecordinginventory, suchasdebitinganinventoryaccountforproductAwhentheaccountfor productBshouldhavebeendebited.Thesedifferencesbetweencountandthe accountingrecordsappearlargefornormalshrinkage,especiallyforcomputer chipsandpokerchips.Wewoulddoublecheckthephysicalcountforerrorsand checktheaccountingrecordsforerrors.Ifnonearefound,thenthecompany mighthaveamajorproblemwithinventorytheftorobsolescence. Solutions 26 2.24 (Justintimemethodsandbackflushcosting.) JournalEntries: (1) CostofGoodsSold........................................................ AccountsPayableMaterials................................ AccountsPayableOtherManufacturing Costs................................................................... WagesPayable........................................................ Torecordcostsofproduction. (2) FinishedGoodsInventory............................................ CostofGoodsSold.................................................. Torecordinventory. 80,000 50,000 20,000 10,000 16,000a 16,000 Taccounts: Accountsand WagesPayableAccounts CostofGoodsSold 50,000 20,000 10,000 80,000 64,000 16,000 FinishedGoodsInventory 16,000 a$16,000=400unitsat$40perunit.($40=$80,000/2,000units.) 27 Solutions 2.25 (MemoryBank;justintimemethodsandbackflushcosting.) JournalEntries: (1) CostofGoodsSold........................................................ AccountsPayableMaterials................................ AccountsPayableOtherManufacturing Costs................................................................... WagesPayable........................................................ Torecordcostsofproduction. (2) FinishedGoodsInventory............................................ CostofGoodsSold.................................................. Torecordinventory. 48,000 26,000 14,000 8,000 4,000a 4,000 Taccounts: Accountsand WagesPayableAccounts CostofGoodsSold 26,000 14,000 8,000 48,000 44,000 4,000 FinishedGoodsInventory 4,000 a$4,000=100unitsat$40.00perunit.($40.00=$48,000/1,200units.) Solutions 28 2.26 (Jobcostsinaserviceorganization.) a. JournalEntries: (1) (2) WorkinProcessSpringsteenProduc tions................................................................ WorkinProcessRCIRecords........................... DirectLaborUnbillable.................................... WagesPayable................................................. 240,000 120,000 24,000 WorkinProcessSpringsteenProduc tions................................................................ WorkinProcessRCIRecords........................... Overhead(Applied)......................................... 80,000 40,000 (3) 140,000 (4) MarketingandAdministrativeCosts WagesandAccountsPayable........................ 20,000 (5a) AccountsReceivable Revenue.......................................................... 600,000 (5b) CostofServicesBilled WorkinProcessSpringsteenProduc tions............................................................ WorkinProcessRCIRecords....................... b. Overhead WagesandAccountsPayable.......................... 384,000 120,000 480,000 140,000 20,000 600,000 320,000 160,000 LOOMISANDASSOCIATES IncomeStatement FortheMonthEndingJanuary31 RevenuefromServices.................................................................. $ 600,000 LessCostofServicesBilled........................................................... 480,000 GrossMargin................................................................................. $ 120,000 Less: DirectLaborUnbillable.......................................................... 24,000 OverheadUnderapplied......................................................... 20,000a MarketingandAdministrative................................................. 20,000 OperatingProfit............................................................................. $ 56,000 a$140,000actual$120,000applied. 29 Solutions 2.27 (Jobcostsinaserviceorganization.) a. JournalEntries: (1) WorkinProcessMountainViewCompany....... WorkinProcessPalatineProductions............... DirectLaborUnbillable...................................... WagesPayable.................................................. 120,000 72,000 8,000 (2) WorkinProcessMountainViewCompany....... WorkinProcessPalatineProductions............... Overhead(Applied)........................................... 90,000 54,000 (3) Overhead............................................................... Variousaccounts(e.g.,Wagesand AccountsPayable)............................................. 140,000 (4) MarketingandAdministrativeCosts................... Variousaccounts(e.g.,Wagesand AccountsPayable)............................................. 60,000 (5a) (5b) b. 200,000 144,000 140,000 60,000 AccountsReceivable......................................... Revenue($200,000fromPalatineand and$100,000fromMountainView)........ 300,000 CostofServicesBilled...................................... WorkinProcessMountainViewCom pany.......................................................... WorkinProcessPalatineProduc tions.......................................................... 336,000 300,000 210,000 126,000 INTERNETDESIGNS IncomeStatement FortheMonthEndingNovember30 RevenuefromServices................................................................ $ 300,000 LessCostofServicesBilled......................................................... 336,000 GrossMargin................................................................................ $ (36,000) LessDirectLaborUnbillable.................................................... 8,000 PlusOverheadOverapplied...................................................... 4,000a LessMarketingandAdministrativeExpense............................. 60,000 OperatingProfit(Loss)................................................................. $ (100,000) a$4,000 = $144,000 applied to jobs and expensed as part of the cost of servicesbilled$140,000actualoverheadincurred. Solutions 210 2.27continued c.Bothjobsshowlossesasfollows: MountainView:Grossmargin=$200,000$210,000=$(10,000). PalatineProductions:Grossmargin=$100,000$126,000=$(26,000). Granted,thereissomeoverappliedoverheadthatincreasesthecostofservices billed,butnotenoughtocreatethenegativegrossmargins. 211 Solutions 2.28 (Jobcostsinaserviceorganization.) a. Wagesand WorkinProcess: Costof AccountsPayable EGadgets ServicesBilled 160,000 100,000 40,000 (1) (1) (3) (2) (4) 90,000 54,000 144,000 (5b) (5b) 224,000 WorkinProcess: Marketingand Overhead EShop AdministrativeCosts (3) 100,000 84,000 (2) (1) (2) 30,000 18,000 48,000 (5b) (4) 40,000 WorkinProcess: AccountsReceivable EFood Revenues (5a) 280,000 (1) (2) 20,000 12,000 32,000 (5b) 280,000(5a) DirectLabor Unbillable (1) 20,000 Entries: (1) (2) (3) (4) (5) b. Laborcostsat$100perhour. Overheadat$60perbillablehour. OverheadactuallyincurredinJune. Marketingandadministrativecosts. Servicesbilled. COMPUTERSYSTEMS,INC. IncomeStatement FortheMonthEndingJune30 RevenuefromServices............................................................................ $ 280,000 LessCostofServicesBilled...................................................................... 224,000 GrossMargin............................................................................................ $ 56,000 Less: DirectLaborUnbillable.................................................................... 20,000 OverheadUnderapplied.................................................................... 16,000a MarketingandAdministrative........................................................... 40,000 OperatingProfit(Loss)............................................................................. $ (20,000) Solutions 212 a$100,000actual$84,000applied. 213 Solutions 2.29 (Jobcostsinaserviceorganization.) a. Wagesand WorkinProcess: AccountsPayable FranklinGroceries 165,000 70,000 20,000 (1) (1) (3) (2) (4) 50,000 20,000 Costof ServicesBilled 70,000 (5b) (5b) 210,000 WorkinProcess: Marketingand Overhead TrumanTrust AdministrativeCosts (3) 70,000 60,000 (2) (1) 100,000 (2) 40,000 140,000 (5b) (4) 20,000 DirectLabor AccountsReceivable Unbillable Revenues (5a)300,000 Entries: (1) (2) (3) (4) (5) b. (1) 15,000 300,000(5a) Laborcostsat$50perhour. Overheadat$20perbillablehour. OverheadactuallyincurredinMarch. Marketingandadministrativecosts. (5a) Franklin Groceries billed for $100,000 and Truman Trust billed for $200,000.(5b)Costofservicesbilled:Franklin$70,000;Truman$140,000 CRAFTYIDEAS IncomeStatement FortheMonthEndingMarch31 RevenuefromServices............................................................................. $ 300,000 LessCostofServicesBilled...................................................................... 210,000 GrossMargin............................................................................................ $ 90,000 Less: DirectLaborUnbillable.................................................................... 15,000 OverheadUnderapplied.................................................................... 10,000a MarketingandAdministrative........................................................... 20,000 OperatingProfit........................................................................................ $ 45,000 a$10,000 = $70,000 actual overhead incurred $60,000 applied to jobs and expensedaspartofthecostofservicesbilled. c.Franklinhasagrossmarginof$30,000andTrumanhasagrossmarginof $60,000.Theratioofgrossmargintorevenueisthesame(30%)forboth,sothey Solutions 214 appearequallyprofitable.Ifwehadtochoosebetweenthetwo,wewouldchoose Trumanbecauseitgeneratesthehighesttotalgrossmargin 2.30 (Appendix2.1)(Computingequivalentunits.) ToCompleteBeginningInventory:[(1.0.60)X 60,000Units)]........................................................... StartedandCompleted.................................................. InEndingInventory:.30X40,000Units....................... Total ..................................................................... 2.31 24,000E.U. 160,000E.U. 12,000E.U. 196,000E.U. (Appendix2.1)(Computingproductcostswithincompleteproducts.) Unitstoaccountfor: BeginningWIP Started&Completed EndingWIP Total Coststobeaccountedfor: BeginningWIP CurrentPeriodCosts Totalcoststobe accountedfor PhysicalUnits 60,000 160,000 40,000 260,000 %Completed DuringPeriod 40% 100% 30% Equivalent Units 24,000 160,000 12,000 196,000 $ 80,000 300,000 $ 380,000 Costper Unit $1.53061perE.U. CostperE.U.done thisperiod$300,000196,000E.U. Costsassignedtounitstransferredout: CostsfrombeginningWIP Currentcostsaddedtocomplete beginningWIP($1.53061X24,000E.U.) Currentcostsofunitsstarted&completed ($1.53061X160,000) Totalcoststransferredout 244,898 $ 361,633 CostsassignedtoendingWIP: ($1.53061X12,000E.U.) $ 18,367 Totalcostsaccountedfor: $ 380,000 215 $ 80,000 36,735 Solutions 2.32 (Actualcostsandnormalcosts.) a. ActualCosts DirectMaterials.......................................................................... DirectLabor................................................................................ VariableManufacturingOverhead............................................ FixedManufacturingOverhead................................................. TotalCost.................................................................................... b. NormalCosts DirectMaterials.......................................................................... DirectLabor................................................................................ VariableManufacturingOverhead............................................ FixedManufacturingOverhead................................................. TotalCost.................................................................................... a$18,000=200%X$9,000. b$27,000=300%X$9,000. 2.33 (Appliedoverheadinabank.) a. Totaloverheadapplied. Quarter 1st 2nd 3rd 4th b. NormalOverhead 200millionX$0.01= $ 2,000,000 200millionX$0.01= $ 2,000,000 200millionX$0.01= $ 2,000,000 100millionX$0.01= $ 1,000,000 EstimatedoverheadfortheYear: 800millionX$0.0l=$8,000,000 Solutions 216 $ 5,000 9,000 20,000 26,000 $ 60,000 $ 5,000 9,000 18,000a 27,000b $ 59,000 2.34 (Analyzingcostsinajobcompany.) Wilson AccountBalanceasofFebruary1: DirectMaterials DirectLabor Overhead $ 3,000 12,000 9,600 $ 24,600 Baker $ 2,400 9,600 7,680 $ 19,680 AddedinFebruary: DirectMaterials DirectLabor Overhead $ 0 3,600 2,880 $ 6,480 $ 31,080 TotalCostofJob Ottley $ 0 6,000 4,800 $ 10,800 $ 30,480 $ 18,000a 4,800 3,840 $ 26,640 $ 26,640 aThedirectmaterialscanbefoundusingthebasiccostflowmodel. BB + TI = TO + EB $6,000 + $12,000 = TO + $0 TO = $18,000 2.35 (Jobcostingforthemovies.) a. Carrying flops in inventory causes studios to report overstated assets. Writingdownthefloptoitsmarketvaluewilldecreasebothinventoryin theassetsectionofthebalancesheet,andprofitsintheincomestatement asthecostsofthefilmarenolongerheldininventorybutexpensed. b. Inventoryshouldbereportedatthelowerofcostormarket.Wheneverthe marketvalueofaproductisknowntobebelowitscost,theproductshould be written down to its market value. The amount of the writedown is expensedintheperiodincurred. 217 Solutions 2.36 (Comparingjobcoststomanagementsexpectations.) Job478: BeginningBalance Labor OverheadApplied Total JobCosts ManagementsExpectations $ 19,600 300 (=.6x$300) 180 $ 20,080 $20,000 Comment:Thetotaljobcostisasclosetomanagementsexpectationsasone canexpect. Job479: BeginningBalance Labor OverheadApplied Total JobCosts ManagementsExpectations $ 9,400 2,600 1,560(=.6x$2,600) $ 13,560 $13,000 Comment: The$560 overrunis(morethan) 4%ofthe expectedjobcosts. Although4%isasmallpercentoftheexpectedtotaljobcosts,itislikelya largepercentofcompanyprofits. Imaginethattheprofitsonthisjobare 10%ofjobcosts.Thenthecostoverrunis40%(=4%/10%)ofthoseprofits. Although apparently small, this cost overrun warrants examination and correctioninfuturejobs. Job480: BeginningBalance Materials Labor OverheadApplied Total JobCosts ManagementsExpectations $ 5,000 600 7,800 4,680(=.6x$7,800) $ 18,080 $15,000 Comment: The jobs cost overrun is 21% of managements expected costs. Thisisamajorproblemthatmustbecorrectedinfuturejobs.. Job481: Materials Labor OverheadApplied Total JobCosts ManagementsExpectations $ 4,200 5,900 3,540(=.6x$5,900) $ 13,640 $10,000 Comment: Cost overrun is 36% of managements expected costs. (This problemisevenbiggerthantheoverrunonJob480.) Solutions 218 2.36continued Job482: Materials Labor OverheadApplied Total JobCosts ManagementsExpectations $ 1,020(=.6x$1,700) 2,600 1,700 $ 5,320 $4,000 Comment: Thecostoverrunis34%ofmanagements expectedcosts. (See commentforJob481.) Note that actual overhead for the month ($12,000) is higher than overhead appliedof$10,980(=$180+$1,560+$4,680+$3,540+$1,020).Therefore,an adjusting entry must be made to the overhead account to transfer the underappliedoverheadamounttocostofgoodssoldasfollows: Dr.CostofGoodsSold.................................................. Cr.Overhead............................................................... ($12,000$10,980=$1,020underapplied) 1,020 1,020 Itisimportanttonotetherelationofcoststoworkdone.Ifbothcostsandwork doneexceedexpectations,thentheCompanymightnothaveaproblem. The Augustdataappeartosupportmanagementsconcerns,however. Aspartofitsefforttoreducecosts,managementshouldidentifyoverhead costdrivers;thatis,thosethingsthatcauseoverheadcosts.Itisunlikelythat laboristheonlycostdriverforoverhead,forexample.Also,examinematerials costs.Materialscostsinconstructionarevolatile.Augustishurricaneseason. Have there been storms that caused destruction which required a lot of materialstorebuildbuildingsandinfrastructure? 219 Solutions 2.37 (Analyzingcostsinajobcompany.) Reed Family Thompson AccountBalanceasofJune1: DirectMaterials DirectLabor Overhead $ 1,000 5,000 5,000 $ 11,000 $ 800 3,200 3,200 $ 7,200 AddedinJune: DirectMaterials DirectLabor Overhead $ 0 1,200 1,200 $ 2,400 $ 13,400 TotalCostofJob Sparks $ 0 2,000 2,000 $ 4,000 $ 11,200 $ 6,000a 1,600 1,600 $ 9,200 $ 9,200 aThedirectmaterialsusedinJunecanbefoundusingthebasiccostflowmodel. BB + TI = TO + EB $2,000 + $4,000 = TO + $0 TO = $6,000 Solutions 220 2.38 (Comparingjobcoststomanagementsexpectations.) Job15: BeginningBalance Labor OverheadApplied Total JobCosts ManagementsExpectations $ 45,000 13,000 7,800(=.6x$13,000) $ 65,800 $70,000 Comment: The job cost is lower than managements expectations. This appearstobegoodnews;managementshouldfindthecauseofthedifference andusethisinformationforfuturebiddingpurposes. Job16: BeginningBalance Labor OverheadApplied Total JobCosts ManagementsExpectations $ 18,000 8,500 5,100(=.6x$8,500) $ 31,600 $30,000 Comment:Costoverrunis5%ofmanagementsexpectedcosts.Investigation ofoverrunswouldbeprudentbecausethiscouldstillbealargepercentofthe expectedprofitsonthejob. Forexample,ifthecompanymakesaprofitof 10%ofexpectedjobcosts,thenthis5%overrunis50%ofthat10%profit. Job17: BeginningBalance Materials Labor OverheadApplied Total JobCosts ManagementsExpectations $ 37,000 23,000 10,500 6,300(=.6x$10,500) $ 76,800 $60,000 Comment: Costoverrunis28%ofmanagementsexpectedcosts. Thisisa significant percentage requiring further investigation and correction to preventsuchoverrunsonfuturejobs. Job18: Materials Labor OverheadApplied Total JobCosts ManagementsExpectations $ 15,500 26,000 15,600(=.6x$26,000) $ 57,100 $60,000 Comment:Costsareabitlowerthanmanagementsexpectations. 221 Solutions 2.38continued Job19: Materials Labor OverheadApplied Total JobCosts ManagementsExpectations $ 29,000 34,500 20,700(=.6x$34,500) $ 84,200 $80,000 Comment:Costoverrunis5%ofmanagementsexpectedcosts.Seecomment onJob16. Note that actual overhead for the month ($70,352) is higher than overhead appliedof$55,500(=$7,800+$5,100+$6,300+$15,600+$20,700).Therefore, an adjusting entry must be made to the overhead account to transfer the underappliedoverheadamountof$14,852tocostofgoodssoldasfollows: Dr.CostofGoodsSold.................................................. Cr.Overhead............................................................... 14,852 14,852 Thisexcessofactualoverappliedissubstantialandshouldbeinvestigated. Usingappliedoverhead,thecostofjobsisunderstated.Costoverrunsarelikely higheronjobsthanstatedabove. Itisimportanttonotetherelationofcoststoworkdone.Ifbothcostsandwork doneexceedexpectations,thentheCompanymaynothaveaproblem.The Novemberdataappeartosupportmanagementsconcerns,however. Aspartofitsefforttoreducecosts,managementshouldidentifyoverheadcost drivers;thatis,thosethingsthatcauseoverheadcosts.Itisunlikelythatlabor istheonlycostdriverforoverhead,forexample. Solutions 222 2.39 (Comparejustintimetoatraditionalaccountingsystem.) Taccountswillshowthefollowingentries. a. BackflushCosting WagesandAccounts WorkinProcess PayableAccounts CostofGoodsSold Inventory 500,000 250,000 800,000 0 1,550,000465,000 (M) (L) (O.H.) 155,000* 155,000 FinishedGoods Inventory 0 310,000* 310,000 *$155,000=.10x$1,550,000and$310,000=.20x$1,550,000. b.TraditionalCostingSystem WagesandAccounts WorkinProcess PayableAccounts MaterialsInventory Inventory 500,000 250,000 800,000 (M) (L) (O.H.) 0 500,000500,000 FinishedGoods Inventory (M) 0 500,000 (L) 250,0001,395,000 (O.H.) 800,000 155,000 CostofGoodsSold 0 1,395,000 1,085,000 310,000 1,085,000 223 Solutions 2.40 (Comparejustintimetoatraditionalaccountingsystem.) a. BackflushCosting WagesandAccounts WorkinProcess PayableAccounts CostofGoodsSold Inventory 100,000 45,000 75,000 220,00055,000 (M) (L) (O.H.) 0 11,000* 11,000 FinishedGoods Inventory 0 44,000* 44,000 *$11,000=.05x$220,000and$44,000=.20x$220,000. b. TraditionalCostingSystem WagesandAccounts WorkinProcess PayableAccounts MaterialsInventory Inventory 100,000 (M) 45,000 (L) 75,000 (O.H.) 0 100,000100,000 0 (M) 100,000 (L) 45,000 209,000 (O.H.) 75,000 11,000 FinishedGoods Inventory 0 209,000 165,000 44,000 Solutions 224 CostofGoodsSold 165,000 2.41 (Appendix2.1) (SanchezCompany;computingequivalentunitsand costflowsunderprocesscosting.) Physical Units Accountingforunits: BeginningWIP 20,000 EndingWIP Total CostsperE.U. thisperiod: Totalcoststransferredout $ 70,000 40% $2,959,00098,000 12,000 100% $ 296,000 2,959,000 $ 3,255,000 Costsassignedtounitstransferredout: CostsfrombeginningWIP Currentcostsaddedtocomplete beginningWIP($30.194X12,000) Currentcostsofunitsstarted&completed ($30.194X70,000) Equivalent Units 60% 70,000a 40,000b 130,000 Started&completed Accountingforcosts: BeginningWIP Currentperiodcosts Total %Completed DuringPeriod 16,000 98,000 Costper Unit $30.194perE.U. 296,000 362,328 2,113,580 $ 2,771,908 $30.80d CostsassignedtoendingWIP: ($30.194X16,000) $ 483,104 Totalcostsaccountedfor: $ 3,255,012c a70,000units=90,000completed20,000completedfrombeginning inventory. b40,000EndingWIP=20,000BeginningInventory+110,000startedin September90,000completed. c$12differenceduetorounding. d$30.80=$2,771,908(20,000units+70,000units). 225 Solutions 2.42 (Appendix2.1)(Equivalentunits;solvingforunknowns.) a. Physical Units TocompleteBeginningInventory 1,000 UnitsStarted&Completed....... 4,000 EndingInventory....................... 3,000 % Completed thisPeriod 40%(2) 100% 40% Equivalent Units 400(1) 4,000 1,200 5,600 (1)400=5,6004,0001,200. (2) 40%=400 1,000. Therefore, BeginningInventorymusthavebeen 60%complete. b. Equivalent Units TocompleteBeginningInventory....................... 4,200 UnitsStarted&Completed................................. 6,000 EndingInventory................................................. 2,000 TotalPeriodConversionCosts........................ Conversion Costs $ 18,270(1) 26,100(2) 8,700 $ 53,070 First, compute unit conversion costs from what we know about ending inventory:$8,7002,000E.U.=$4.35perE.U. (1)$18,270=4,200X$4.35. (2)$26,100=6,000X$4.35. Solutions 226 2.43 (AlamedaCompany;completingmissingdata.) Theanswersappear(onthefollowingpage)asfootnotestotheseTaccounts. AccountsPayable 6,000 4/1 40,000 42,000b 8,000 4/30 DirectMaterialsInventory 4/1 12,000 42,000b 43,000f 4/30 WorkinProcessInventory 4/1 4,500 43,000f 11,000g 89,000 Overhead Actual Applied 14,800 15,600c FinishedGoodsInventory 4/1 11,000 d 84,000 89,000 4/30 16,000 CostofGoodsSold 15,600c 800e WagesPayable 31,200 84,000 d 31,200 a 4/30 5,300 (=2,600hrs. X$12) Under/OverappliedOverhead e 800 227 Solutions 2.43continued. Note:PredeterminedOverheadRate =$180,000/30,000directlaborhours =$6perDirectLaborHour. aWorkinProcessInventory: DirectMaterials................................................................................. DirectLabor($12X150hours).......................................................... Overhead($6X150hours)................................................................. Total............................................................................................... $ 2,600 1,800 900 $ 5,300 bDirectMaterialsPurchased =$40,000+$8,000$6,000 =$42,000. cOverheadApplied =$6X2,600directlaborhours =$15,600. dCostofGoodsSold =$11,000+$89,000$16,000 =$84,000. eOverappliedOverhead =$15,600$14,800 =$800.(Note:CreditmaybetocostofGoodsSold insteadofUnder/OverappliedOverhead.) fDirectMaterialsUsed =$89,000+$5,300$4,500$15,600$31,200 =$43,000. gEndingDirectMaterialsInventory =$42,000+$12,000$43,000 =$11,000. Solutions 228 2.44 (PremierPrinting,Inc.;incompletedatajobcosting.) The following information should be included (in summary) in a report to management. WorkinProcess CostofGoodsSold Cash JobNo.11 JobNo.11 M 4,000* 4,000 L 19,200*38,400 O3 9,600 19,200 6/1 32,800 L1 19,200 O4 9,600 0 9,200* M L O2 4,000* 38,400 19,200 61,600* WagesPayable JobNo.12 JobNo.12 M5 6,000 6,000 L6 24,00024,000 O7 12,00012,000 0 64,000* M L O 6,000 24,000 12,000 42,000 Overhead Overhead JobNo.13 Adjustment Actual Applied 40,000*32,000 8,000 10 9 M 3,200* L 20,800* O8 10,400 6/30 34,400 8,00010 Note:Seefootnotesonfollowingpage. 229 Solutions 2.44continued. FootnotestoProblem2.44. Mreferstodirectmaterials. Lreferstodirectlabor. Oreferstomanufacturingoverhead. *Numbersgivenintheproblem. 1 Labortocompletejobis$19,200sincethebeginninginventorywas50%complete. 2 Appliedoverhead =$61,600$4,000materials$38,400directlabor. =$19,200 Appliedoverhead =$19,200/$38,400=0.50timeslabordollars. 3 Overheadinbeginninginventory=0.50X$19,200=$9,600. 4 OverheadappliedinJune=0.50X$19,200=$9,600. 5 MaterialsforJobNo.12 =PurchasesmaterialsforJobNo.13 =$9,200$3,200 =$6,000 6 LaborforJobNo.12 =TotaldirectlaborcostsLaborforJobNo.11Labor forJobNo.13 =$64,000$19,200$20,800 =$24,000 7 OverheadforJobNo.12=0.50X$24,000=$12,000. 8 OverheadforJobNo.13=0.50X$20,800=$10,400. 9 AppliedOverhead 10 =$9,600+$12,000+$10,400=$32,000. UnderappliedOverhead =ActualApplied =$40,000$32,000=$8,000. Solutions 230 2.45 (MidwestInsuranceCompany;evaluatingcostsystemsusedinfinancialservice companies.) a. Mr. Frank's decision regarding the most appropriate type of accounting system actually involves two decisions which are somewhat independent. Midwestcurrentlyusestheequivalentofajobordersystembasedonactual costs,whichisbeingcomparedwithNorthern'ssystem,effectivelyaprocess costsystem(withinthethreecategoriesofloans)basedonstan 231 Solutions 2.45a.continued. dardcosts.Thedecisionstobemadetheninvolvethefollowingdimensions: 1. Thedegreetowhichaccountingdataisaccumulatedtoa specificunitof outputasopposedtogroupsofunits(jobordervs.processcost),and 2. Thedegreetowhichactualcostdataisusedtosatisfytherequirements oftheusersofaccountingdataasopposedtousing predetermineddatato satisfytheserequirements. Inpractice,thenatureofthebusinesswillsuggestthetypeofsystemtobe used, but the decisions are by no means absolutemany different combinationsandvariationscananddoexist. Severalspecificcharacteristicsoftheproduct(service)andbusinessare helpfulinguidingthedecisionbetweenpotentialsystems.Theseinclude: 1. Homogeneityofoutputstandarddataandprocesscostingaregenerally betterinsituationswhereoutputisrelativelyuniform. 2. Nature/knowledge of demandin order to use predetermined rates (especially overhead), volume and mix of demand must be somewhat predictable. 3. Performance measurementstandard costs provide a (hopefully realistic)benchmarkforevaluatingperformance,butactualdatamust becollectedtoallowcomparison. 4. Impactofvarianceswherevariancescanbesignificantandmorethan temporary, the additional detail provided in actual and joborder systemsmaywarranttheiruse. 5. Estimating/billingwhere this must be performed for individual customers,detaileddataareusuallyrequired. Summarizingthesetwosystemsintermsoftheadvantages/disadvantages, Midwest's system provides detailed data which may be beneficial in performancemeasurementandprojectionofcostchangesifthemixofloan types varies in the future, with the disadvantage of additional record keepingandclericalcosts.Northern'ssystemhastheobviousadvantageof beinginexpensiveandprovidingastandardforperformancemeasurement, but maynot providethedetail tomake this measurement meaningful or allow effective variance analysis. In addition, Northern's overhead allocationschememayrelyonarelativelyconsistentmixofloantypes. Solutions 232 2.45a.continued. Althoughtheabovecharacteristicswouldsuggest achoiceofsystems givenequalcostsoftheaccountingsystems,thecost/benefitquestionmust beaddressedduetothefactthatcostsoftheaccountingsystemsare not equal.Theadditionalrecordkeepingandclericalcostsinherentinthejob order/actualsystem,whicharecurrentlyofprimaryconcerntoMr.Frank, shouldnotbeincurredunlesstheyarejustifiedbythebenefitsprovidedby greatercostvisibility. In the case of the Northern and Midwest systems, the following differences between the two are important in making the choice of an appropriatesystem: Both the average cost and the variability of Midwest's loan processingcostsaremuchgreaterthanthoseofNorthern. The components of Midwest's loan processing costs can be significantly different from loan to loan (e.g., some loans involve travelcostswhileothersdonot). Midwestapparentlyreceivesloansonanindividualbasisratherthan asapackage. Midwest's processing costs include outside services such as consulting over which it may have very little control or ability to predict. Inlightoftheforegoing,Midwestshouldprobablystayforthemostpart withitspresentsystemofusingactualcosts,sincethevariabilityinitsloan processing costs would make it very difficult to develop meaningful standardsinthefirstplace.Itwouldbeverydifficulttoestimateinadvance theprocessingcostsforaspecificloan,andperformancemeasurementbased onstandardcostsmaynotcarrymuchmeaning, especiallyintheareaof outsideservices.ItwouldalsobedifficultforMidwesttoallocateoverhead costs, since not all loans entail the same overhead cost components and there does not seem to be a rational basis for allocation of these costs. Midwest could,however,batchsomeoftheloanstogetheriftherewereno specific"needtoknow"thecostofprocessingaparticularloan. Midwest probablylooksatoverallloanprocessingcostsindevelopinganinterestrate orfeechargedtobanksandotherfinancialinstitutions,and,ifso,should notbeoverlyconcernedwithprocessingcostsonindividualloans.Although Midwest'sMortgageDivisionmanagerpointstovariabilityasanargument against standards, he does not mention any benefit related to collecting costsforeachloan. 233 Solutions 2.45continued. b. Exhibit1showsthecostflowsundertherespectivesystems. Underboth, totalcostsareinitiallyaccumulatedinvariousPayablesaccounts. Under Northern'ssystem,a"product"isaloancategory,withaDirectLaborand anOverheadaccountmaintainedundereachcategory. Costsarerecorded intheotheraccounts(otherthanPayables)atstandard,withthedifferences collectedinVarianceaccounts. Thus,Northernmaintainsatotalofeight accounts exclusive of Payables. Midwest, of course, does not require Variance accounts, but maintains up to six accounts for each loan and, hence,mustmaintainasignificantlygreaternumberoftotalaccounts. Exhibit1 ComparativeCostFlows Standard/ProcessCosting NorthernInsurance VariousPayables FHA/VALoans 58,172 (1) DirectLabor Overhead 74,626 (2) (1) 4,200 (2) 5,460 ConventionalLoans DirectLabor Overhead (1) 31,160 (2) 40,508 DevelopmentLoans DirectLabor Overhead (1) 20,440 (2) 26,572 Direct LaborVariance (1) 2,372 OverheadVariance (2) 2,086 Solutions 234 235 Solutions 2.45b.continued. Exhibit1(cont.) Actual/JobOrderCosting MidwestInsurance VariousPayables Loan#A4810136 47,291(1) DirectLabor Telephone Travel 4,843 (2) (1) 1,184 (2) 113 (3) 415 2,739 (3) 11,800 (4) 9,950 (5) Appraisal Legal Other 1,470(6) (4) 1,500 Loan#A4811237 DirectLabor Telephone Travel (1) 3,631 (2) 42 Appraisal Legal Other (4) 2,300 Loan#B4219361 DirectLabor Telephone Travel (1) 814 (2) 78 Appraisal Legal Other (5) 1,500 (6) 150 Loan#C3921341 DirectLabor Telephone Travel (1) 4,191 (2) 240 (3) 110 Appraisal Legal Other (5) 2,200 Note: Debitsdonotequalcreditsforeachentrybecausethisisonlyapartiallistingof loansprocessedinJulyasshowninthetextExhibit2.12. Solutions 236 2.46 (OklahomaHardware,Inc.;reconstructmissingdata.) Thisisachallengingproblem.Weputtheworkinprocessaccountontheboard forthebigpicture,thensolveforeachitemintheaccountasfollows: WorkinProcess BeginningBalance (a) 86,200 53,500 (d) Transferredto DirectMaterials (b) 70,314 FinishedGoods DirectLabor (c) 67,700 204,014 (f) DisasterLoss OverheadApplied (e) 33,300 EndingBalance 0 The calculations are shown below. We usually present these using both T accountsandthefollowingformulas. (a) Given. (b) DirectMaterials =BeginningInventory+PurchasesEnding InventoryIndirectMaterials a a a =$49,000 +$66,400*$43,000 $2,086 = $70,314 *Purchases =AccountPayable,Ending+CashPayments AccountsPayable,Beginning a a a =$50,100 +$37,900 $21,600 =$66,400 (c) DirectLabor =PayrollIndirectLabor a a =$82,400 $14,700 =$67,700 (d) Cost Transferred to Finished Goods = Finished Goods, Ending + Cost of GoodsSoldFinishedGoods,Beginning a a a =$37,500 +($396,600 $348,600 )$32,000 a =$53,500 (e) OverheadApplied =EndingManufacturingOverheadBeginning ManufacturingOverhead+OverappliedOverhead a a a =$217,000 $184,900 +$1,200 =$33,300 (f) Loss a =$86,200 +$70,314+$67,700+$33,300 $53,500 =$204,014 Note:Theinsurancecompanymaydisputepayingthe$1,200 overappliedoverhead. a Givenintheproblem. 237 Solutions 2.47 (Jobcostingandethics.) a. ItwouldbeunethicalforAndretofalsifyjobcostreportsby improperlyassigningcoststotheCanadiangovernmentjob whichwereactuallypartofthecostoftheGeneralElectric job. SinceAndresbosssuggestedthiscourseofaction,he should approach higher levels of management with the problem. Given the potential illegality and other possible negative ramifications of this problem (such as lost reputation),itislikelythatmanagementwilldecidetowrite offthecostoverrunsinsteadoffalselyreportingthem. b. The fact that Andres company is reimbursed on the Canadiangovernmentcontractmakesitparticularlyenticing tochargetheexcesscoststothisproject.However,sincethe Canadiangovernmentcontractisbasedoncosts,itmaybe an illegal action for the company to misrepresent costs charged to this project. If this action is discovered and provenincourt,thecompanycouldbeliablefortheexcess charges,interestandpunitivedamages.Andreandhisboss couldbeheldresponsibleforcivilandcriminalpenaltiesnot tomentionthelossoftheirjobsandtheirreputations. 2.48 (ComparisonofjustintimeintheU.S.andJapan.) Japanese companies have been at the forefront in utilizing justintime techniques(particularlyintheautomobileindustry),andthereforehavemore experiencewithJITthanU.S.companies. Japanalsohaslimitedresourcesin terms of land and storage, which leads to higher storage costs than for U.S. companies. Further, transportationdistances aregreater intheU.S.making reliable delivery more difficult. Thus, it is not surprising that the chemical industryinJapanismoreeffectivelyimplementingjustintimetechniques. 2.49 (CusterManufacturing;jobcostingusingequivalentunits.) a. Cost of WIP inventory on December 31, Year 1 for four departments: FabricatingDepartment Plastic $12.75per Square Foot Costper Equivalent Unit Direct Materials $12,750/100 =$127.50 CostofEndingInventory: Solutions 238 Direct Labor $1,424/89 =$16 Overhead $.45X855/95 =$4.05 ($127.50X12)+($16X6)+($4.05X12)+($12.75X50)=$2,312.10 239 Solutions 2.49a.continued. TestingDepartment Costper Equivalent Unit Transferin Costs $127.50+$16+$4.05 =$147.55 Direct Labor $444/74 =$6 Overhead $301.92/74 =$4.08 CostofEndingInventory: ($147.55X7)+($6X7)+($4.08X7)=$1,103.41 AssemblyDepartment Transferin Costs Costper $147.55+$6 Equivalent +$4.08 Unit =$157.63 Frames $408.52 Direct Labor $612/51 =$12 Overhead $232.56/51 =$4.56 CostofEndingInventory: ($157.63X18)+($408.52X31)+($12X8)+($4.56X8)=$15,633.94 ShippingDepartment Transferin Costs Costper $157.63+$408.52 Equivalent +12+$4.56 Unit =$582.71 Packing Material $75 Direct Labor $256/32 =$8 Overhead $64/32 =$2 CostofEndingInventory: ($582.71X19)+($75X16)+($8X8)+($2X8)=$12,351.49 b. CostofGoodsSoldfor23UnitsShipped: ($582.71X23)+($75X23)+($8X23)+($2X23)=$15,357.33 c. CostofUnitsSpoiled: TestingDepartment ShippingDepartment ($147.55X15)+($6X6)+($4.08X6)+($582.71X1)+($75X1)+($8X1)+($2X1) =$2,941.44 Solutions 240 2.49dcontinued d.Thecostofunitssold=$688(=$15,357/23units),whichislessthanthetarget of$700perunit.Thespoilageof$2,941.44isgreaterthan10%ofthecostofgoods sold(CGS=$15,357).Socostsofgoodunitsareslightlylowerthantarget(whichis good)andthecostofspoilageisgreaterthantarget(whichisbad).Eveniftheunit hadnotbeendroppedinshipping,thespoilagecostswouldhavebeengreaterthan 10%ofcostofgoodssold.Spoilageappearstobeanareaofconcern. 241 Solutions
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NYU - ACCT - 202
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Stanford - EEAP - ee359
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Chapter 21. Pr = Pt 103 = Pt 103 = Pt Gl 4d 4 10 4 1002 = c/fc = 0.062 Pt = 4.39KW2 Pt = 438.65KWAttenuation is very high for high frequencies 2. d= 100m ht = 10m hr = 2m delay spread = = 3. =2 (x +xl) x+x l c= 1.33x +xl =(ht + hr )2 + d2 (ht
Stanford - EEAP - ee359
Chapter 31. d = vt2r + r = d + 2hdEquivalent low-pass channel impulse response is given byc(, t) = 0 (t)ej0 (t) ( 0 (t) + 1 (t)ej1 (t) ( 1 (t)G0 (t) = 4d l with d = vt0 (t) = 2fc 0 (t) D00 (t) = d/cD0 = t 2fD0 (t)dtvfD0 (t) = cos 0 (t)0 (t)
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Chapter 41. C = B log2 1 +C=SN0 Blog2 1+ NSB1B0As B by LHospitals ruleC=S1N0 ln 22. B = 50 MHzP = 10 mWN0 = 2 109 W/HzN = N0 BC = 6.87 Mbps.Pnew = 20 mW, C = 13.15 Mbps (for x1, log(1 + x) x)B = 100 MHz, Notice that both the bandwidth
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Chapter 61. (a) For sinc pulse, B =12Ts Ts =12B= 5 105 sP(b) SN R = N0bB = 10Since 4-QAM is multilevel signallingPEs2sSN R = N0bB = N0 BTs = NEBB Ts = 120E SNR per symbol = Ns = 50ESNR per bit = Nb = 2.5 (a symbol has 2 bits in 4QAM)
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Chapter 71. Ps = 103QPSK, Ps = 2Q( s ) 103 , s 0 = 10.8276.MPout (0 ) =1e 0ii=1 1 = 10, 2 = 31.6228, 3 = 100.M =10Pout = 1 e 1= 0.6613M =20Pout = 1 e 11e 0M =30Pout = 1 e 11e 0= 0.191721e2 03= 0.0197M 1e/2. p ( ) = M 1 e/
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Chapter 101. (a)(AAH )T= (AH )T .ATT= (AT ) AT= AAH (AAH )HFor AAH ,= AAH = , i.e. eigen-values are realAAH = QQH(b) X H AAH X = (X H A)(X H A)H = X H A 0 AAH is positive semidenite.(c) IM + AAH = IM + QQH = Q(I + )QHAH positive semidenite
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Chapter 111. See Fig 12B = 100 KHzfc-Bfc+Bfc = 100 MHzFigure 1: Band of interest.B = 50 KHz, fc = 100 MHzHeq (f ) =1=fH (f )Noise PSD = N0 W/Hz. Using this we getfc +BNoise Power ==N0 |Heq (f )|2 dffc Bfc +BN0f 2 dffc B3 (fc +B )= N
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Chapter 151. City has 10 macro-cellseach cell has 100 users total number of users = 1000Cells are of size 1 sqkmmaximumdistance traveled to traverse = 2km2 time = 30 = 169.7sIn the new setupnumber of cells = 105 microcellstotal number of users =
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TABLE OF CONTENTSPROBABILITY THEORYLecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 Lecture 8 Lecture 9 Lecture 10 Lecture 11 Lecture 12 Lecture 13 Basics Independence and Bernoulli Trials Random Variables Binomial Random Variable A
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2. Independence and Bernoulli Trials (Euler, Ramanujan and Bernoulli Numbers)Independence: Events A and B are independent ifP ( AB ) = P ( A) P ( B ).(2-1) It is easy to show that A, B independent implies A, B; A, B ; A, B are all independent pairs. F
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3. Random VariablesLet (, F, P) be a probability model for an experiment, and X a function that maps every , to a unique point x R, the set of real numbers. Since the outcome is not certain, so is the value X ( ) = x . Thus if B is some subset of R, we m
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4. Binomial Random Variable Approximations, Conditional Probability Density Functions and Stirlings FormulaLet X represent a Binomial r.v as in (3-42). Then from (2-30) n k nk P (k1 X k 2 ) = Pn ( k ) = p q . k = k1 k = k1 k k2 k2(4-1)Since the binom
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5. Functions of a Random VariableLet X be a r.v defined on the model (, F , P ), and suppose g(x) is a function of the variable x. DefineY = g ( X ).(5-1)Is Y necessarily a r.v? If so what is its PDF FY ( y ), pdf fY ( y ) ? Clearly if Y is a r.v, the
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6. Mean, Variance, Moments and Characteristic FunctionsFor a r.v X, its p.d.f f X ( x) represents complete information about it, and for any Borel set B on the x-axisP ( X ( ) B ) =Bf X ( x ) dx .(6-1)Note that f X ( x) represents very detailed info
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7. Two Random VariablesIn many experiments, the observations are expressible not as a single quantity, but as a family of quantities. For example to record the height and weight of each person in a community or the number of people and the total income i
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8. One Function of Two Random VariablesGiven two random variables X and Y and a function g(x,y), we form a new random variable Z asZ = g ( X , Y ).(8-1)Given the joint p.d.f f XY ( x , y ), how does one obtain f Z ( z ), the p.d.f of Z ? Problems of t
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9. Two Functions of Two Random VariablesIn the spirit of the previous lecture, let us look at an immediate generalization: Suppose X and Y are two random variables with joint p.d.f f XY ( x, y). Given two functions g ( x, y ) and h( x, y ), define the ne
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10. Joint Moments and Joint Characteristic FunctionsFollowing section 6, in this section we shall introduce various parameters to compactly represent the information contained in the joint p.d.f of two r.vs. Given two r.vs X and Y and a function g ( x, y