前導(dǎo)班定量資深培訓(xùn)師

1、FRM一級(jí)培訓(xùn)項(xiàng)目iveysis講師：么崢資深培訓(xùn)師地點(diǎn)： 么崢么崢稱：FRM，浙江大學(xué)數(shù)學(xué)學(xué)士，浙江大學(xué)金融學(xué)。教授課程：數(shù)量分析，與產(chǎn)品，估值與風(fēng)險(xiǎn)模型工作背景：曾就職于交通風(fēng)險(xiǎn)管理部，負(fù)責(zé)全行新資本協(xié)議的實(shí)施工作；參與各新資本協(xié)議有關(guān)項(xiàng)目，包含信用風(fēng)險(xiǎn)初級(jí)法改造、市場(chǎng)風(fēng)險(xiǎn)驗(yàn)證、第二支柱建設(shè)等項(xiàng)目；跟進(jìn)三定量測(cè)算與最新動(dòng)態(tài)?，F(xiàn)就職于某制總行風(fēng)險(xiǎn)部，負(fù)責(zé)模型驗(yàn)證工作。：96 2 -96 2-iveysis 20%Discrete and continuous probability distributionsPo ulation and sam le sisticsSistical infe

2、rence and hypothesis testingEstimating the parameters of distributionsGraphical represenion of sistical relationshipsLinear regreswith sin le and multi le re ressorsThe ordinary least squares (OLS) methodreting and using regresHypothesis testing andcoefficients ,the t-servalsistic, and other outputH

3、eteroskedasticity and multicollinearitySimulation methodsEstimating correlation and volatility: EWMA and GARVolatility term structuress96 3 -96 3-Readings foriveysis10. Michael Miller, Mathematics and Sistics for Finanl RiskManagement (Hoboken, NJ: John Wiley & Sons, 2012).Chapter 2 -ProbabilitiesCh

4、apter 3 -Basic SisticsChapter 4 -DistributionsChapter 5 -Hypothesis Testing &ervals11. James stock and Mark Watson,roduction to econometrics, Briefedition(ton: Pearson Education, 2008).Chapter 4 - Linear regreswith one regressorsingle regressor: Hypothesis Tests andChapter 5 - RegreservalsChapter 6

5、-Linear regreswiwith multiple regressorsChapter 7 - Hypothesis Tests and regreservalsultiple96 4 -96 4-Readings foriveysis12. Dessislava Pachamanova and FrFabozzi, Simulation andOptimization in Finance (Hoboken, NJ: John Wiley & Sons, 2010)Chapter 4 Simulation Ming13.John Hull, Options, Futures, and

6、 Other Derivatives, 8th Edition ork: Prentice Hall, 2012).(New YChapter 22 Estimating Volatilities and Correlations如何學(xué)好定量分析部分？抓住基本概念搞清來龍去脈忽略理論推導(dǎo)多做習(xí)題練習(xí)96 5 -96 5-FRMive OutlinePreparationThe Usage of FinanBasic conceptsl ComputerNumerical Characteristics of Random VariablesProbability Distributions (

7、Discrete & Continuous)PoEstimate andervalsHypothesis TestsLinear regresMonte Carlo Methods96 6 -96 6-FRMive OutlinePreparationTheerest rateEffective annual rateSummation and differentiationConcave and convexityPV、FV、PMT、I/YNPV、IRR96 7 -96 7-Theerest rateHow much is $100 after 3 years if y=10%?FV=？20

8、13$100How much is the present value of $100 three years later if y=10%?PV3=FV3/(1+y)3PV2=FV2/(1+y)2PV1=FV1/(1+y)FV11FV22FV33096 8 -96 8-Effective Annual Rate (EAR)Simple v.s. compoundingCompounding conventions are important ideterminingffeiannual rate (EAR), in order to compare securities with diffe

9、rent compounding periods, we must convert their yields to EARTeralized formula to compute the EARr nEAR=1+ n 1n : number of compounding periods per yearr : annual rate (quoted)Annually, semiannually, quarterly, monthly, continuously compoundingkly, daily,n , EAR erContinuously compounded rate196 9 -

10、96 9-Summation and differentiationSummation NoionsionThe Summatininn X . Xn2ii1i1i1XProperties of the Summation Operatorn k nki1 kXi k Xi( Xi Yi ) XiYi(a bXi ) na b Xi96-96 10-10Summation and differentiationf(x)y x) f ( x0f ( x0 )f ( x ) l lim0 x 0f (x0 )y0 y f (yf ( x x) f ( x )00 xxy f (x )00 x0 x

11、0 x9161-96x0 x)Summation and differentiationf ( x0 x) f ( x0 )f ( x ) lim0 x x) x 0f ( x0f ( x0 ) lim(x)2x 096-96 12-12PV、FV、PMT、I/YNPV, IRRPV, PMT, FV, I/YTCt(1 y)ty)Tt 1Example1: calculate the price of a bond. Coupon rate=5%, 10 years, annually compounding. What about semi-annual compounding?Examp

12、le2: the price of a bond is 99. Coupon rate=5%, 10 years, annually compounding.Calculate the yield of the bond.NPV, IRRExample3: a project is going to earn 10, 20, 20 million dollars IRR is 10%. Calculate the NPV.Example4: a project is going to earn 10, 20, 20 million dollars invests 35 million doll

13、ars. Calculate the IRR.hree years. Supe thehree years. A company96-96 13-13FRMive OutlinePreparationThe Usage of FinanBasic conceptsl ComputerNumerical Characteristics of Random VariablesProbability Distributions (Discrete & Continuous)PoEstimate andervalsHypothesis TestsLinear regresMonte Carlo Met

14、hods96-96 14-14指定金融計(jì)算器的使用BA II Plus/Profes常見al96-96 15-15計(jì)算器基礎(chǔ)TI BAII PLUS計(jì)算器的基本設(shè)定主要功能按鍵: 都印在鍵上。如按右上方【ON/OFF】 鍵，表示開機(jī)關(guān)機(jī)。次要功能按鍵:按【2ND】切換鍵之后，顯示寫在按鍵上方的次要功能。如【2ND 】【ENTER】表示調(diào)用SET功能。小數(shù)位數(shù)的設(shè)置:默認(rèn)為兩位小數(shù)；更改設(shè)置時(shí),依次按【2ND】【】，表示調(diào)用 FORMAT功能，出現(xiàn)DEC=2.00，若要改為四位小數(shù)，輸入4，再按【ENTER】，出現(xiàn)DEC=4.0000。時(shí)一般最好設(shè)為4位小數(shù)。這樣輸入金額時(shí)可以萬元計(jì)，結(jié)果的小數(shù)

15、點(diǎn)4可以精確到元。位，小數(shù)位數(shù)設(shè)置將保持有效，不會(huì)因退出或重新開機(jī)而改變，要重新設(shè)置FORMAT才會(huì)改變。數(shù)字重新輸入按【CE/C】鍵。96-96 16-16計(jì)算器基礎(chǔ)特殊計(jì)算功能操作負(fù)號(hào)功能（-2： 【2】【+|-】）括號(hào)的使用1/X 功能（1/2：【2】【1/X】）ex功能（e2：【2】【2nd】【ex】）yx功能（23 ：【2】【 yx】 【3】 【=】）（21/3：【2】【 yx】 【3】 【1/X】 【=】）nCr功能（C3：【10】 【2nd】【nCr】【3】 【=】）10nPr功能（ P3：【10】 【2nd】 【nPr】【3】 【=】）10階乘功能（5！=120 【5】【2nd

16、】【 】 ）96-96 17-17貨幣時(shí)間價(jià)值時(shí)間價(jià)值 / 財(cái)務(wù)計(jì)算【CPT】【N】【I/Y】【PV】:計(jì)算( Compute )供款期數(shù)( Number of Payments )利率(erest Rate )現(xiàn)在價(jià)值供款( Premium / Payment )將來價(jià)值( Future Value )【PMT】【FV】【2ND】【 CLR TVM】: 清除全部貨幣時(shí)間值(All Clear )PV 、FV、 N 、I/Y、 PMT這五個(gè)貨幣時(shí)間價(jià)值功能鍵中會(huì)存有上次運(yùn)算的結(jié)果，通過【OFF】或【CE/C】鍵無法清除其中數(shù)據(jù)。正確的清空方法是按【2ND】調(diào)用【CLR TVM 】。在計(jì)算器中輸

17、入 I/Y 時(shí)，不需要加百分號(hào)，例如： I/Y 8%，直接輸入 8 【I/Y】 即可。為表述簡單，凡直接書寫第二功能鍵，即表示先按2ND，然后按其所對(duì)應(yīng)的主功能鍵。96-96 18-18貨幣時(shí)間價(jià)值運(yùn)算規(guī)則：PV現(xiàn)值、FV終值、PMT年金、I/Y利率、N期數(shù)，運(yùn)用財(cái)務(wù)計(jì)算器計(jì)算貨幣時(shí)間價(jià)值的五大變量。只要輸入任何四個(gè)變量，可以求出剩下一個(gè)變量。例題一：由現(xiàn)值求終值投資100元，以累積率為10%，投資期限為10年，問這項(xiàng)投資10年后一共可？計(jì)算器按鍵依次為：10【N】；10【I/Y】；0【PMT】；-100【PV】；【CPT】【FV】。計(jì)算結(jié)果為：FV=259.3742例題二：由終值求現(xiàn)值面值1

18、00元的零息債券，到期收益率為6%，10年到期，該債券當(dāng)前的價(jià)?格應(yīng)該是計(jì)算器按鍵依次為：10【N】；6【I/Y】；0【PMT】；100【FV】；【CPT】【PV】。計(jì)算結(jié)果為PV=-55.839596-96 19-19NP求VIRN和RPVIRR例題B公司計(jì)劃以100一臺(tái)新機(jī)器，這家公司希望的投資回報(bào)率為10%，未來五年內(nèi)公司預(yù)計(jì)現(xiàn)金流如下表所示，試求凈現(xiàn)值和內(nèi)部收益率。023451-100203020202096-96 20-20年數(shù)預(yù)計(jì)現(xiàn)金流120230320420520現(xiàn)金流現(xiàn)金流方法一96-96 21-21按鍵解釋顯示CF 2ND CLR WORK清除CF功能中的CF0=0.0000

19、100+/-ENTER期初投入CF0=-100.0000 20 ENTER第一期現(xiàn)金流C01=20.0000 30 ENTER第二期現(xiàn)金流C02=30.0000 20 ENTER第三期現(xiàn)金流C03=20.0000 20 ENTER第四期現(xiàn)金流C04=20.0000 20 ENTER第五期現(xiàn)金流C05=20.0000NPV 10 ENTER折現(xiàn)率10%I=10.0000計(jì)算NPVNPV=-15.9198IRR CPT計(jì)算IRRIRR=3.3675現(xiàn)金流現(xiàn)金流方法二96-96 22-22按鍵解釋顯示CF 2ND CLR WORK清除CF功能中的CF0=0.0000100+/-ENTER期初投入CF

20、0=-100.000 20 ENTER第一期現(xiàn)金流C01=20.0000 30 ENTER第二期現(xiàn)金流C02=30.0000 20 ENTER第三期現(xiàn)金流C03=20.0000 3 ENTER現(xiàn)金流20將會(huì)連續(xù)出現(xiàn)三次F03=3.0000NPV 10 ENTER折現(xiàn)率10%I=10.0000 CPT計(jì)算NPVNPV=-15.9198IRR CPT計(jì)算IRRIRR=3.3675統(tǒng)計(jì)運(yùn)算統(tǒng)計(jì)運(yùn)算例題已知某之前五年的收益率結(jié)果為：0%，5%，10%，15%，20%。求它的均值和方差。操作步驟：DATA功能96-96 23-23按鍵解釋顯示【2nd】【7】進(jìn)入DATA功能X010.0000【2nd】【

21、CE/C】-【CLR WORK】清除DATA功能中的X010.00000【ENTER】第一個(gè)收益率X01=0.0000【】【】5【ENTER】第二個(gè)收益率X02=5.0000【】【】10【ENTER】第三個(gè)收益率X03=10.0000【】【】15【ENTER】第四個(gè)收益率X04=15.0000【】【】20【ENTER】第五個(gè)收益率X05=20.0000統(tǒng)計(jì)運(yùn)算統(tǒng)計(jì)運(yùn)算S功能另一只密切相關(guān)，五年對(duì)應(yīng)收益率為1%，4%，10%與該，13%，21%。求回歸直線。96-96 24-24按鍵解釋顯示【2nd】【DATA】進(jìn)入DATA功能X010.0000【2nd】【CE/C】-【CLR WORK】清除D

22、ATA功能中的X010.00000【ENTER】【】1【ENTER】【】第一個(gè)收益率X01=0.0000 Y01=1.00005【ENTER】【】4【ENTER】【】第二個(gè)收益率X02=5.0000 Y01=4.000010【ENTER】【】10【ENTER】【】第三個(gè)收益率X03=10.0000 Y01=10.000015【ENTER】【】13【ENTER】【】第四個(gè)收益率X04=15.0000 Y01=13.000020【ENTER】【】21【ENTER】【】第五個(gè)收益率X05=20.0000 Y01=21.0000【2nd】【8】-【S】LIN表示線性關(guān)系LIN【】【】【】【】各統(tǒng)計(jì)與回

23、歸指標(biāo)結(jié)果n=5.0000 計(jì)算日期計(jì)算日期計(jì)算日期間隔功能例：2012年1月12日到2012年3月15日【2nd】【1】月.日年： 【1】【.】【1212】【ENTER】【】月.日年： 【3】【.】【1512】【ENTER】【】【CPT】96-96 25-25FRMive OutlinePreparationThe Usage of FinanBasic conceptsl ComputerNumerical Characteristics of Random VariablesProbability Distributions (Discrete & Continuous)PoEstima

24、te andervalsHypothesis TestsLinear regresMonte Carlo Methods96-96 26-26FRMive OutlineBasic conceptsRandom events, results and eventsPopulation and sampleRandom variablesProbability and probability calculationProbability density function and cumulative density function96-96 27-27Random events, result

25、s and eventsPhenomenonCertain phenomenonUncertain phenomenonRandom Experiment & Random Variables: is an uncertainty/number.Sam le S ace or Po ulationSample PoRandom Event: is a singlee or a set ofes.Mutually exclusive events: are events same time.Collectively exhaustive events: are those est cannot

26、both happen at thet include allsible96-96 28-28Random events, results and eventsRandom VariablesRandom variables are denoted by capital letters X, Y, Z, etcThe values taken by these variables are often denoted by small letters,x, y, z, etc. eDiscrete random variableContinuous random variableProbabil

27、ityProbability of an Event: The Classical or A Priori DefinitionP( A) number ofes favorable to Atotal number ofes96-96 29-29Probability and probability calculationProperties of ProbabilitiesThe probability of an event alwayss betn 0 and 1. Thus, the probabilityof event A, P(A), satisfies this relati

28、onship:0 P( A) 1If A,B,Care mutually exclusive events, the probabilityt any one ofthem will occur is equal to the sum of the probabilities of their individualoccurren.P( A B C .) P( A) P(B) P(C) .If A,B,C,are mutually exclusive and collectively exhaustive set of events,the sum of the probabilities o

29、f their individual occurrenis 1.P( A B C .) P( A) P(B) P(C) . 196-96 30-30Probability and probability calculationProperties of ProbabilitiesAddition rule：P( A B) P( A) P(B) P( AB)For every event A, there is an event A, called the complement of A:P A A AA P96-96 31-31Probability and probability calcu

30、lationUnconditional probability: P(A), P(B)Conditionalrobabilit : P A BWe want to find out the probabilityt the event A occurs knowingt the event B has already occurred. This probability is called theconditionalrobabilitof Aiven B.P( A|B) P( AB) ;P(B) 0P(B)The conditional probability of A, given B,

31、is equal to the ratio of theirjoprobability to the marginal probability of B. In like manner,P(B|A) P( AB) ;P( A) 0P( A)probability P(AB)=P(A) P(B|A)=P(B) P(A|B)Jo96-96 32-32Probability and probability calculationIndependent eventsThe occurrence of A has no influence on the occurrence of BB is indep

32、endent of AP(AB)=P(A)P(B) P(B|A) = P(B) P(A|B) = P(A)Three events A1, A2, A3 are independent if Ak ) P(Aj )P(Ak ), j k.P(Ajwhere j, k=1,2,3andP( A1 A2 A3 ) P( A1 )P( A2 )P( A3 )96-96 33-33Probability density function and cumulative density functionRandom Variables and Their Probability Distributions

33、Probability Distribution of a Discrete Random VariableProbability Mass Function (PMF) or Probability Function (PF)f ( X xi ) P( X xi ), i 1, 2, 3.Properties of the PMFFor example: Binomial n=3 p=0.5, x xifBinomial: n=3 p=.50 f (x ) 1xP(x)i01230.1250.3750.3750.1251.000f (x ) 1ix96-96 34-34P(x)0.40.30

34、.20.10.00123C1Probability density function and cumulative density functionProbability Distribution of a Continuous Random VariableProbability density function (PDF)x22 ) f (x)dxx1PA PDF has the following properties:The total area under the curve f(x) is 1P(x1Xx2)is the area under the curvebetPn x1 a

35、nd x2. P P P2 )3.22296-96 35-35Probability density function and cumulative density functionCumulative Distribution Function (CDF)F ( X ) P( X x)F(x)1F bP(a X b)=F(b) - F(a)F(a)0 xf(x)P(a X b) = Area underf(x) betn a and b= F(b) - F(a)xa0b96-96 36-36abFRMive OutlinePreparationThe Usage of FinanBasic

36、conceptsl ComputerNumerical Characteristics of Random VariablesProbability Distributions (Discrete & Continuous)PoEstimate andervalsHypothesis TestsLinear regresMonte Carlo Methods96-96 37-37FRMive OutlineNumerical Characteristics of Random VariablesExpecionsVariance, standard deviationCovarianceCor

37、relation coefficientSkewnesskurtosis96-96 38-38ExpecionsExpected Value: A Measure of Central Tendency themomentE( X ) x f (x) x1P(x1 ) x2 P(x2 ) xn P(xn )XE( X ) xf (x)dxProperties of Expected Value1. If b is a constant, E(b)=b 2. E(X+Y)=E(X)+E(Y)3. In general, E(XY) E(X)E(Y); If X and Y are indepen

38、dent random variables, then E(XY) =E(X)E(Y)4. E(X2) E(X)2If a is a constant, E(aX)=aE(X)If a and b are constants, then E(aX+b)=aE(X)+E(b)=aE(X)+b96-96 39-39Variance and standard deviationVariance: a Measure of DisperThe definition of variance the second momentVAR( X ) x E( X 22X )itive square root o

39、f VAR(X), x , is known as the standardThedeviation.To compute the variance, we use the following formula:VAR( X ) ( X )2 P( X )ixiXiVAR( X ) (x x )f (x)dx2VAR( X ) E( X 2 ) E( X )296-96 40-40Variance and standard deviationProperties of VarianceThe variance of a constant is zero. By definition, a con

40、stan variability.If X and Y are two independent random variables, thens noVAR(X+Y)=VAR (X)+VAR (Y)andVAR (X-Y)=VAR (X)+VAR (Y).b is a constant, then: VAR (X+b)=VAR (X)4. If a is constant, then: VAR (aX)=a2VAR (X).a and b are constant, then: VAR (aX+b)=a2VAR (X)6. If X and Y are independent random va

41、riables and a and b areconstants, then VARaX+bY =a2VARX +b2VARY7. For compuional convenience, we can get: VAR (X)=E(X2)-E(X)2 ,E( X 2 ) x2 f ( X )xt96-96 41-41CovarianceCovariancecov(X, Y)E(X - E(X)(Y - E(Y)E(XY) - E(X)E(Y)Covariance measures how one random variable moves wirandom variable.notherCov

42、ariance ranges from negative infinity toitive infinity.Properties of CovarianceIf X and Y are independent random variables, their covariance is zero.cov(X,Y)=cov(Y,X)3. cov(X, X) E(X-E(X)(X-E(X) 2 (X)cov(a+bX, c+dY) bd cov( X ,YIf X and Y are NOT independent, then:var( X Y ) var X var Y 2 cov X Y96-

43、96 42-42Correlation coefficientCorrelation coefficient cov(X,Y) XYxyProperties of Correlation coefficientCorrelation measures the linear relationship bet variables.n two randomCorrelation has no units, ranges from 1 to +1.If two variables are independent, their covariance is zero, therefore, the cor

44、relation coefficient will be zero. The converse, however, is NOT true.For example, Y=X2Varianof correlated Variables.var( X Y ) var( X ) var(Y ) 2 x y96-96 43-43Correlation coefficient96-96 44-44Correlation coefficientreionr = +1perfectitive correlation0 r +1itive linear correlationr = 0no linear co

45、rrelation1 r 3=30=00Exs kurtosisTails(amingFat tailnormalThailsame variation)96-96 46-46FRMive OutlinePreparationThe Usage of FinanBasic conceptsl ComputerNumerical Characteristics of Random VariablesProbability Distributions (Discrete & Continuous)PoEstimate andervalsHypothesis TestsLinear regresMo

46、nte Carlo Methods96-96 47-47FRMive OutlineProbability Distributions (Discrete & Continuous)BernoulliBinomialNormal distribution96-96 48-48Some Important Probability DistributionsBernoulli random variableP(Y=1)=pP(Y=0)=1-pBinomial random variablethe probability of x suces in n trails n x px)X )x( 1 p

47、nxp(P) x Jacob Bernoulli (1654-1705)數(shù)學(xué)家Expecions and varian96-96 49-49ExpecionVarianceBernoulli random variable (Y)pp(1-p)Binomial random variable (X)npnp(1-p)Some Important Probability DistributionsThe Cumulative Binomial Probability Tablex0.050.7740.9770.9991.0001.0001.0000.10.5900.9190.9911.0001.

48、0001.0000.20.3280.7370.9420.9931.0001.0000.30.1680.5280.8370.9690.9981.0000.40.0780.3370.6830.9130.9901.0000.50.0310.1880.5000.8130.9691.0000.60.0100.0870.3170.6630.9221.0000.70.0020.0310.1630.4720.8321.0000.80.0000.0070.0580.2630.6721.0000.90.0000.0000.0090.0810.4101.0000.950.0000.0000.0010.0230.22

49、61.000012345F (x) P( X x) P(i)all i xF(x1)P(X) = F(x)Deriving Individual Probabilities from Cumulative ProbabilitiesFor example:P(3) F (3) F (2) .813 .500 .31350-95906Some Important Probability DistributionsThe Binomial Distribution- Overviewp = 0.5Binomial Probability: n=4 p=0 5p = 0.1p = 0.3Binomi

50、al Probability: n=4 p=0 0.50.4n = 0.10.0012x34012x34Binomial Probability: n=10 p=0.1Binomial Probability: n=10 p=0 3Binomial Probability: n=10 p=0.50.4n = 0.001 2 3 4 5 6 7 108x9Binomial Probability: n=20 p=0.1Binomial Probability: n=20 p=0.3

51、Binomial Probability: n=20 p=0.50 20 20 2n = 0 00 00 016817 192016817 192016817 1920 xxxBinomial distributionse more symmetric as n increases and as= . .51- -95916P(x)P(x)P(x)P(x)P(x)P(x)P(x)P(x)P(x)0.10.00 1 2 3 4 5 6 7 108 9x0.10.00 1 2 3 4 5 6 7 108 9xBinomial Probabi

52、lity: n=4 p=01234xSome Important Probability DistributionsNormal DistributionAs n increases, teistribuaches Normal Distribution.n = 6Binomial Distribution: n=6, p=.5n = 10Binomial Distribution: n=10, p=.5n = 14Binomial Distribution: n=14, p=.50 30 30 30 20 20 0 0

53、0 000 00123x45612345x671809xNormal Distribution: = 0, =1 Normal Probability Density Function:0.4103 x f (x)exp 02220.100wheree 2 .7182818 3 . and.-505x52- -95926P(x)P(x)P(x)f(x)Some Important Probability DistributionsThe Shof the Normal Distribution Density FunctionThe normal curve is symmetrical Th

54、e two halves are identicalTheoretically, the curve extends to +Theoretically, the curve extends to -The mean, median, and mode are equal.Properties: N (, 2 ) , Fully described by its two parameters and 2.XBell-shd, Symmetrical distribution swness=0; kurtosis=3.A linear combination of two (or more) i

55、ndependent normally distribution random variables is normally distributed.53- -95936Some Important Probability DistributionsTheervalsy 68% of all observations fall y 90% of all observations fall y 95% of all observations fall y 99% of all observations fallApproxima Approxima ApproximaApproximahe he

56、heheerval erval 1.65 erval 1.96 erval 2.5854- -95946Some Important Probability DistributionsThe standard normal distributionN(0,1) or ZStandardization: if XN( , ), then Z X N(0,1)How to use Z-table?How we use the standard normal distribution to compute variousprobabilities?X 75 .9Example: X N (70，9)

57、 , compute the probability ofZ 75.9 70 1.96 , then compute P(Z 1.96) 1 0.975 0.025364.12 X 75.9Question 1: compute the probability ofQuestion 2: compute the probability of 64.12 X and X 75.955- -95956Some Important Probability DistributionsThe central limit theorem (CLT)Laplace: ifis a random sample

58、 from any population (i.e.,nprobability distribution) with mean tends to be normally distributeX2Xand, the sample meanX2t X nsample size increases indefiniX xy (technically, infiniy 30) N (0,1) xn xStandard Error (se) of mean X:nHowever, the populations standard deviation is almost never known.Inste

59、ad,use the standard deviation of the sample mean. 1 X X 2S 2xin 156- -9596Some Important Probability DistributionsThe Chi-Square( 2 ) Probability DistributionnX N (0,1), 2 2 (n)2iXii1(n 1)s2 (n 1)22057- -95976Some Important Probability DistributionsThe t Distribution (Students distribution)Z ( X X )

60、 N (0,Recallt,1) ,bothandare known.2x/nXXe we only know2xSupand estimateby its (sample) estimatorX(Xi - X )2, we obtain a new variable.2Sx =n 1t= X Xt(n1)S /nxX N (0,1);Y 2 (n);X且X,Y獨(dú)立，t t(n)Y n58- -95986Some Important Probability DistributionsProperties of the t DistributionSymmetricThe mean of t d