數(shù)理統(tǒng)計(jì)試驗(yàn)數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第1頁(yè)統(tǒng)計(jì)圖1.直方圖hist(頻數(shù)),ecdfhist(頻率)2.正態(tài)概率圖normplot3.盒圖(箱線圖)boxplot4.QQ圖(分位數(shù)圖)

qqplot5.繪制數(shù)據(jù)最小二乘擬合線lsline數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第2頁(yè)樣本均值mean樣本方差var(x,0)var(x,1)樣本標(biāo)準(zhǔn)差std樣本中心矩moment樣本中位數(shù)median樣本眾數(shù)mode樣本極差range平均絕對(duì)偏差mad樣本偏度skewness樣本峰度kurtosis描述性統(tǒng)計(jì)數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第3頁(yè)累積分布函數(shù)二項(xiàng)分布binocdf泊松分布poisscdf均勻分布unifcdf正態(tài)分布normcdf卡方分布chi2cdft分布tcdfF分布fcdf數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第4頁(yè)其它函數(shù)概率密度函數(shù)normpdf,poisscdf逆累積分布函數(shù)norminv,poissinv隨機(jī)數(shù)生成normrnd,poissrnd離散均勻分布unidrnd,連續(xù)均勻分布unifrnd均值與方差normstat,poisstat經(jīng)驗(yàn)分布函數(shù)圖像ecdf,cdfplot數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第5頁(yè)參數(shù)預(yù)計(jì)單個(gè)總體參數(shù)預(yù)計(jì)值和對(duì)應(yīng)區(qū)間預(yù)計(jì):正態(tài)擬合normfit二項(xiàng)擬合binofit泊松擬合poissfit指數(shù)擬合expfit最大似然預(yù)計(jì)mle數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第6頁(yè)核密度預(yù)計(jì)命令:ksdensity例題:產(chǎn)生60個(gè)來(lái)自由N(0,1)和N(5,1)組成混合正態(tài)樣本，畫(huà)出密度函數(shù)核預(yù)計(jì)圖像。x=[randn(30,1);5+randn(30,1)];[f,xi]=ksdensity(x);plot(xi,f)數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第7頁(yè)[f1,xi1]=ksdensity(x,'kernel','normal');[f2,xi2]=ksdensity(x,'kernel','box');[f3,xi3]=ksdensity(x,'kernel','triangle');[f4,xi4]=ksdensity(x,'kernel','epanechnikov');plot(xi1,f1,'k')holdonplot(xi2,f2,'r:')plot(xi3,f3,'b-')plot(xi4,f4,'c--')legend('Gaussian','Uniform','Triangle','Epanechnikov','Location','NorthWest')數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第8頁(yè)例題分別產(chǎn)生100個(gè)標(biāo)準(zhǔn)正態(tài)分布隨機(jī)數(shù)和參數(shù)為1指數(shù)分布隨機(jī)數(shù)，并畫(huà)出它們正態(tài)概率圖。x=normrnd(0,1,100,1);y=exprnd(1,100,1);normplot(x)normplot(y)數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第9頁(yè)隨機(jī)生成5組9個(gè)整數(shù)數(shù)據(jù)，求每組數(shù)據(jù)中位數(shù)。x=fix(30*rand(9,5));median(x)固定隨機(jī)數(shù)生成結(jié)果:mystream=RandStream('mt19937ar','Seed',0);RandStream.setDefaultStream(mystream);例題數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第10頁(yè)例題產(chǎn)生二項(xiàng)分布b(20,0.75)隨機(jī)數(shù)，并求最大似然預(yù)計(jì)和置信區(qū)間。x=binornd(20,0.75,100,1);[p,pci]=mle('bino',x,0.05,20)p是預(yù)計(jì)量，pci是對(duì)應(yīng)置信區(qū)間。數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第11頁(yè)例題

某批產(chǎn)品中有正品數(shù)a，次品數(shù)ka(k為待估參數(shù))。從中任取一只，若為正品，記X=1，不然記X=0.現(xiàn)在有放回地抽取n次，得m只正品。求k矩預(yù)計(jì)和最大似然預(yù)計(jì)。矩預(yù)計(jì)法clearsymskexmnpx1=1/(1+k);px0=k/(1+k);ex=m/n;k=solve(ex-1*px1-0*px0,k)數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第12頁(yè)最大似然預(yù)計(jì)法clearsymskmnlike=k^(n-m)/(1+k)^n;like1=log(like);like2=diff(like1,k);k=solve(like2,k)

會(huì)用函數(shù)diff和solve求解矩預(yù)計(jì)和最大似然預(yù)計(jì)數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第13頁(yè)練習(xí)產(chǎn)生10個(gè)標(biāo)準(zhǔn)正態(tài)分布隨機(jī)數(shù)，指出分布特征，并畫(huà)出經(jīng)驗(yàn)分布函數(shù)。一組來(lái)自正態(tài)分布總體樣本觀察值683,681,676,678,679,672,求總體均值和標(biāo)準(zhǔn)差點(diǎn)預(yù)計(jì)值及置信水平為0.95置信區(qū)間.數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第14頁(yè)練習(xí)答案1.x=normrnd(0,1,10,1)[h,stats]=cdfplot(x)2.x=[683,681,676,678,679,672];[mu,sigma,muci,sigmaci]=normfit(x)數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第15頁(yè)假設(shè)檢驗(yàn)?zāi)彻S生產(chǎn)10歐電阻，假定電阻值服從正態(tài)分布，標(biāo)準(zhǔn)差為0.1歐。現(xiàn)隨機(jī)地抽取10個(gè)電阻，測(cè)得它們值為:9.9,10.1,10.2,9.7,9.9,9.9,10,10.5,10.1,10.2。問(wèn)能否定為該廠電阻平均值為10歐?(取alpha=0.1)H_0:mu=10，H_1:mu不等于10。標(biāo)準(zhǔn)差已知，用ztest()格式。x=[9.9,10.1,10.2,9.7,9.9,9.9,10,10.5,10.1,10.2];[h,sig,ci]=ztest(x,10,0.1,0.1)h=0，表示不拒絕原假設(shè)。數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第16頁(yè)2.某種元件壽命X(單位:h)服從正態(tài)分布，均值和方差均未知。現(xiàn)隨機(jī)抽取16只元件，測(cè)得其壽命為:159,280,101,212,224,379,179,264,222,362,168,250,149,260,485,170。問(wèn)是否有理由認(rèn)為元件平均壽命大于225h?(取alpha=0.05)H_0:mu<=225，H_1:mu>225。標(biāo)準(zhǔn)差未知，用ttest()格式。x=[159,280,101,212,224,379,179,264,222,362,168,250,149,260,485,170];[h,sig,ci]=ttest(x,225,0.05,1)%tail是'both','right'或'left';或0,1,或-1。默認(rèn)是0.h=0，表示不拒絕原假設(shè)。數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第17頁(yè)3.杜鵑總是把蛋生在別鳥(niǎo)巢中，現(xiàn)在從甲和乙兩種鳥(niǎo)巢分別得到杜鵑蛋9只和15只，測(cè)得杜鵑蛋長(zhǎng)度數(shù)據(jù)以下:甲:21.2,21.6,21.9,22,22,22.2,22.8,22.9,23.2;乙:19.8,20,20.3,20.8,20.9,20.9,21,21,21.1,21.2,21.5,22,21.9,21.1,22.3.假設(shè)這兩個(gè)樣原來(lái)自同方差正態(tài)總體，試判別杜鵑蛋長(zhǎng)度差異是因?yàn)殡S機(jī)原因造成，還是與它們被發(fā)覺(jué)鳥(niǎo)巢不一樣相關(guān)?（alpha=0.05）H_0:mu_1=mu_2，H_1:不等[h,sig,ci]=ttest2(x,y,0.05)h=1表示不能接收原假設(shè)，杜鵑蛋長(zhǎng)度與被發(fā)覺(jué)鳥(niǎo)巢相關(guān)數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第18頁(yè)4.化肥廠用自動(dòng)包裝機(jī)包裝化肥，某日測(cè)得9包化肥重量(單位：kg)以下：49.4,50.5,50.7,51.7,49.8,47.9,49.2,51.4,48.9檢驗(yàn)每包化肥重量方差是否等于1.5?(alpha=0.05)H_0:sigma^2=1.5，H_1:sigma^2不等于1.5[h,p,varci,stats]=vartest(x,1.5,alpha,'both')h＝0，p>apha，接收原假設(shè)。h=0或者p>alpha時(shí)接收原假設(shè)數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第19頁(yè)5.針對(duì)問(wèn)題3中數(shù)據(jù)，假設(shè)總體來(lái)自兩個(gè)正態(tài)分布且均值未知，檢驗(yàn)甲和乙兩種杜鵑蛋長(zhǎng)度方差是否相等?(alpha=0.05)H_0:sigma_1^2=sigma_2^2，H_1:不等[h,p,varci,stats]=vartest2(x,y,alpha,'both')h=0或者p>alpha時(shí)接收原假設(shè)數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第20頁(yè)6.某工廠生產(chǎn)一個(gè)黃金飾品，現(xiàn)隨機(jī)抽取20件，測(cè)得每件含黃金量以下:0.693,0.749,0.654,0.670,0.662,0.672,0.615,0.606,0.690,0.628,0.668,0.611,0.606,0.609,0.601,0.553,0.570,0.844,0.576,0.933試驗(yàn)證這些飾品含金量是否服從正態(tài)分布?(alpha=0.05)[H,P,LSTAT,CV]=lillietest(x)H=1,P<0.05，拒絕原假設(shè)，檢驗(yàn)統(tǒng)計(jì)量LSTAT大于臨界值CV，一樣拒絕原假設(shè)。數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第21頁(yè)Lilliefors檢驗(yàn)(當(dāng)總體均值和方差未知時(shí)，用對(duì)應(yīng)樣本均值和樣本方差去代替，然后使用Kolmogorov-Smirnov檢驗(yàn))，檢驗(yàn)樣本是否服從指定分布。lillietest函數(shù)可用分布有正態(tài)分布、指數(shù)分布和極值分布，不能用于非位置尺度分布族分布檢驗(yàn)。數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第22頁(yè)7.其它檢驗(yàn)方法:卡方擬合優(yōu)度檢驗(yàn)，檢驗(yàn)樣本是否服從指定分布：chi2gof,Jarque-Bera檢驗(yàn)，檢驗(yàn)樣本是否服從正態(tài)分布：jbtest,單個(gè)樣本Kolmogorov-Smirnov檢驗(yàn)，檢驗(yàn)樣本是否服從指定分布：kstest兩個(gè)樣本Kolmogorov-Smirnov檢驗(yàn)，檢驗(yàn)兩個(gè)樣本是否服從相同分布：kstest2數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第23頁(yè)8.某醫(yī)院外科用兩種手術(shù)方法治療基本情況相同肝癌患者11例。患者采取隨機(jī)方法分配到不一樣手術(shù)組。每例手術(shù)后生存月數(shù)以下，試比較兩種手術(shù)方法術(shù)后生存月數(shù)是否有差異?(alpha=0.05)甲：2,3,5,6,12乙：3,7,8,8,10,11采取Wilcoxon秩和檢驗(yàn)[p,h]=ranksum(x,y,0.05)H=0,p>0.05,接收原假設(shè)，沒(méi)有顯著差異數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第24頁(yè)練習(xí)產(chǎn)生100個(gè)正態(tài)隨機(jī)數(shù)N(20,5^2)樣本，分別在方差已知和未知情形下，檢驗(yàn)總體均值mu=20和mu=23.5。(取alpha=0.05)數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第25頁(yè)練習(xí)答案x=normrnd(20,5,1,100);總體方差已知時(shí)用ztest()格式，未知時(shí)用ttest()格式[hz0,sigmaz0,ciz0]=ztest(x,20,5)[hz1,sigmaz1,ciz1]=ztest(x,23.5,5)[ht0,sigmat0,cit0]=ttest(x,20)[ht1,sigmat1,cit1]=ttest(x,23.5)結(jié)果表明:兩種情況下都接收假設(shè)mu=20且拒絕假設(shè)mu=23.5數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第26頁(yè)回歸分析對(duì)以下數(shù)據(jù)，求出因變量y、自變量x一元線性回歸模型，并對(duì)各參數(shù)進(jìn)行檢驗(yàn)。x=[100,110,120,130,140,150,160,170,180,190]';y=[45,51,54,61,66,70,74,78,85,89]';plot(x,y,'r.')%觀察散點(diǎn)圖X=[ones(10,1),[100,110,120,130,140,150,160,170,180,190]'];[b,bint,r,rint,stats]=regress(y,X)數(shù)理統(tǒng)計(jì)實(shí)驗(yàn)lastedition第27頁(yè)b是回歸方程中對(duì)應(yīng)參數(shù)預(yù)計(jì)值，bint是回歸方程對(duì)應(yīng)參數(shù)95％置信區(qū)間，r和rint分別表示殘差及殘差對(duì)應(yīng)95％置信區(qū)間。stats輸出四個(gè)數(shù)字分別表示相關(guān)系數(shù)R^2、F統(tǒng)計(jì)量觀察值、檢驗(yàn)p值和誤差方差預(yù)計(jì)值。所得回歸方程為y=-2.73