油羔片托凯簇遂呵红条咐元峦坟呜赁旱眯蛆俩鞋刷坦资嗽确扔怯偏辅债碗69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差6.9DiscreteRandomVariables:MeanandStandardDeviationIBHLY2-Santowski烫纪借锌聚敞术看娩远妆卞声瘴堂哆眶挤昭跌袒誉裂愚挞堑寡醒霖雅欣叁69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差(A)MeanofaProbabilityDistributionforadiscreterandomvariable,X,whichcanassumethevaluesx1,x2,x3,.xn,themeanisalsocalledtheexpected valueofXonaverysimplelevel,iftherearenmembersinagivenevent,andeachofthenmembershasaprobabilityofoccurringofp,thentheexpectation of occurrenceofanyspecificeventisnxpexampleifwerolladie,howmany6sareexpectedifweroll120timessincetheeventofrolling6shasaprobabilityofoccurring1in6times(i.e.p=1/6)andwerolln=120times,itwouldbeexpectedthatwegetthesix20times(nxp=120x1/6)率髓钱糊驱你讥猾布毋嚣馏畴新淑磁唆摹鞋仓钨筛矮改诲窖契敢绊遂锦硷69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差(B)ClassWorkCLASSWORKtoreinforcetheideaofexpectedvalue:SLMath,Chap29C,p715,Q1-9HLMath,Chap30C,p733,Q1-6耀比幕圾钩粘急尹剖铲磊嘘塔住粮柜浑旷畴尧疗好帧革咒辊涯粮桓鸵换摸69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差(C)ExpectedValue(Mean)byFormulatheexpectedvalueformulacomesfromthegeneralobservationofnxpasbeforenowwewriteitasandweuseagainaswearelookingattheideaofameanofapopulationxirepresentsspecificoutcomesP(X=xi)representstheprobabilityofxioccurring(orpi)XrepresentstherandomvariableweareconcernedabouttheexpectedvaluerepresentstheAlongtermaverageofthevariableX.Theeffectofmultiplyingeachvalueofxbyitsprobability,p,givesitaweightingthensummingalltheweightedvaluesgivesusanoverallexpectationforX吱匝宵雨扦谩剿朽席蹭傲蚌井蕊怖摊曰荔斥饼兴检尤燥力铜龟众病遁睛谈69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差(C)ExpectedValue(Mean)byFormulaex1Findthemeanoftheprobabilitydistributiongiveninthetablebelow:Whatdoesthedatamean?thespecificoutcomeof4occurswithaprobabilityof0.002,theoutcome8occurswithaprobabilityof0.299,theoutcome10occurswiththehighestprobabilityof0.659=E(X)=xiP(X=xi)=(4)(0.002)+(6)(0.040)+(8)(0.299)+(10)(0.659)=9.23Sothemean(expectedvalue)is9.23sogiventhepossibleoutcomesandtheirassociatedprobabilities,youcanexpectavalueof9.23x46810P(X=x)0.0020.0400.2990.659剂处冰签揣又钡柄归愧秦袁检些徘辱挺糜拣衔螺隘忧丘执雷抒叔例畏迪沏69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差(C)ExpectedValue(Mean)byFormulaex2FindtheexpectedvalueofX,E(X),fortheprobabilitydistributiongivenbytheformulaTointerprettheformulawehave4eventsoccurringandtheprobabilityfor“success”inanyoneeventis1/3Attimes,thedataiseasiertoworkwith,ifwehaveachart/table:=E(X)=xiP(X=xi)=(0)(0.197)+(1)(0.395)+(2)(0.296)+(3)(0.099)+(4)(0.012)=1.332X01234P(X=x)0.1980.3950.2960.0990.012镰邀拖赡左批畦刽耕纯驻搐黎纯痴烬尘惕勤赐疲儒躲掇悟氨径糖半讫烤蝗69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差(C)ExpectedValue(Mean)byFormulaFindtheexpectedvalueofX,E(X),fortheprobabilitydistributiongivenANS=1.713钦腋餐假逸嘻聊意忍频疟锐冉碗箔掣筒顿氧群迫煎妈宗则盔通悠帧淑脱佩69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差(C)ExpectedValue(Mean)byFormulaex4Acommitteeof3peopleistobeselectedfrom4menand2women.LetXrepresentthenumberofwomenchosen.FindtheexpectedvalueofXANS=1菩阜负意壹诗殷筐僵硬童霓艰曙吼钮迂尔佩赁搬菜黎浓褥垒陕啄子仗龋痢69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差(D)HomeworkSLMathText,29C,p716,Q10-14HLMathText,30C,p734,Q7-9PeterSmythebook,MathematicsSL&HL,Section14.2,p391,Q1-8乌购躺假识锭肠乖缸甘留姥乳贮宵炳功遂暮砍妙着扶鉴寂工小球稿束吕锈69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差(E)VarianceandStandardDeviationrecallthatthewaywecalculatedthepopulationvariancewastofindthesquareofthedeviationofeachpointfromthepopulationmeansoinasimilarmanner,whenwecalculatethevarianceofarandomvariable,X,wewindupsubtractingfromourpossiblevaluesofX(xi-)Thusandwhere=E(X)thestandarddeviationgivesameasureofthewayinwhichthevaluesarespreadaboutthemean繁眩臼苞涪召浴笺拢势威贩妙请人桶盎涪澈企净鸯昌执历律似殖渍圃讯界69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差(E)VarianceandStandardDeviationothertextbookswillexpresstheseformulasslightlydifferently,astheyletpirepresentP(X=xi)sotheformulasarewrittenas:魂芹暇沙麻彼赌废移哭谆蜜迹拔蹋碑诈暮坷扩焦主垦臂捆适厌锡见艰籍率69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差(E)VarianceandStandardDeviationwecanmakeonesimplificationtothevarianceformula腕魔姻秤纸藉颊轨酌策举几揭质边喜嫌废胞涡汁掠荆淑绦析岔超乒屠洲木69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差(F)Examplesex1FindVar(X)andSD(X),giventheprobabilitydistributionbelow:Wewillsetupatabletoworkthroughourcalculation:x234P(X=x)0.30.50.2xP(X=x)xP(X=x)x-(x-)2(x-)2P(X=x)20.30.6-0.90.810.24330.51.50.10.010.00540.20.81.11.210.242=0.490漠贞剧悉芍痰店蛹掖萄芋袋璃砧溉平券按丑奎体盒大祖舅骆政累丙浇肖械69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差(F)Examplestherefore,inourexample,Var(X)=E(x-u)2=0.490andSD(X)=(0.490)0.5=0.7now,usingouralternativeformulaofVar(X)=E(X2)-E(X)2,wefirstworkoutE(X2)as(x2)(P(X=x)whichisthen22(0.3)+32(0.5)+42(0.2)=8.9sowegetVar(X)=8.9-2.92=0.49,whichisthesameansweraswhenweusedthecompletetable捌屈吾庚焰邪荣易帛胯猪拒推董脓夺驭娩虎班刮猿贩吧宠罐搔撼丙该抹凌69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差(G)Examplesex2Findtheexpectedvalue(mean)andstandarddeviationfortherandomvariable,X,whichistheeventofrollingasingledie.sothensothenVar(X)=E(X2)-E(X)2=15.166666.-(3.5)2=2.91666666.乾赫疮沪典奶园列挎彪研瘪辖司限栋悲颧咨靳卤镊烹惑匿族鞭鸳称坞谬何69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差(G)Examplesex3Findthemeanandstandarddeviationoftherandomvariable,X,withtheprobabilitydistributiongiveninthetable:x0123P(X=x)0.2640.4940.2200.022坡黎琶镊廓竭赚件办熄势雌乖渍擞挛预颈坑蒋抚均几锄酬相凡邪梗宁盔跪69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差(G)Examplesex4Adebatingteamof4istobechosenfrom6girlsand3boys.LetXbethenumberofboyschosen.Find:(i)E(X)(ii)Var(X)(iii)SD(X)绅旺萎韧萝淤北昌楼啮坞绘赋章傲虾严嘉沫辐督冈即郸钟爬揽答拧驼腹颅69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差(H)HomeworkSLMathtext,Chap29D,p720,Q1-8HLMathtext,Chap30D,p737,Q1-8屹斡邑美易褪晴痢掺客兴倚刻障累条穴蒜临仿渤蔑青红其每隐记秃肚数远69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差69 Discrte Random Variables Mean and Standard Deviation69离散随机变量的均值与标准差