常用統(tǒng)計(jì)分布實(shí)用教案

1、分布的概率密度為分布的概率密度為定理定理)(4 . 52n 其它其它00)2(21)(2122xexnxpxnn.)(2圖圖分布的概率密度曲線如分布的概率密度曲線如n 第1頁/共40頁第一頁，共41頁。分布的性質(zhì)分布的性質(zhì)2 性質(zhì)性質(zhì)(xngzh)1(xngzh)1).(,),(),(2122221222122221221nnnn 則則立立獨(dú)獨(dú)并且并且設(shè)設(shè))(2分布的可加性分布的可加性 (此性質(zhì)可以(ky)推廣到多個(gè)隨機(jī)變量的情形).(,), 2, 1(),(21212222mmiiiiinnnmin 則則獨(dú)立獨(dú)立相互相互并且并且設(shè)設(shè)第2頁/共40頁第二頁，共41頁。性質(zhì)性質(zhì)(xngzh)2(

2、xngzh)2.2)(,)(),(2222nDnEn 則則若若證明證明(zhngmng),1, 0( NXi因?yàn)橐驗(yàn)? 1)()(2 iiXDXE所以所以2242)()()(iiiXEXEXD )(2分布的數(shù)學(xué)期望和方差分布的數(shù)學(xué)期望和方差 dxexXExi24422123221xedxdxexexxx22232221321第3頁/共40頁第三頁，共41頁。 niiXEE122)( 故故 niiXDD122)( niiXE12)(,n niiXD12)(.2n )(2iXD, 213 ., 2, 1ni 22213xedx3dxexexx22222133第4頁/共40頁第四頁，共41頁。性質(zhì)性

3、質(zhì)3dtexnnPxntxnn22212lim,),(222有則對(duì)任意設(shè)且獨(dú)立同分布因而獨(dú)立且每個(gè)其中由假設(shè)和定義證明,),1 , 0(, 6 . 52222121122ninniiXXXNXXXXX), 2 , 1(2)(, 1)(22niXDXEii 第5頁/共40頁第五頁，共41頁。2lim2xnnPn由中心極限定理得xniindtexnnXPt2221lim12).2 ,().1 , 0(2,222nnNNnnn近似進(jìn)而近似服從很大時(shí)當(dāng)也即分布分布的極限分布是正態(tài)即第6頁/共40頁第六頁，共41頁。.)()(,)1 , 0(,12265432221121621分分布布服服從從使使得得求

4、求樣樣本本的的一一組組為為來來自自正正態(tài)態(tài)總總體體設(shè)設(shè)例例 XXXXCXXCYCCNXXX )1 , 0(2),2 , 0(2121NXXNXX 則則解解),(),(1044065436543NXXXXNXXXX則則同理221XX 且且46543XXXX與相互獨(dú)立第7頁/共40頁第七頁，共41頁。2212)(XX 所以所以)2()4(226543 XXXX .41,2121 CC則則第8頁/共40頁第八頁，共41頁。).(,/,),(),(.ntTtnnYXTYXnYNX記為分布的服從自由度為則稱隨機(jī)變量獨(dú)立且設(shè)定義21075 tntnnnthn,1221)(212 分布的概率密度函數(shù)為分布的

5、概率密度函數(shù)為)(nt分布分布t2.第9頁/共40頁第九頁，共41頁。圖圖分布的概率密度曲線如分布的概率密度曲線如t.0對(duì)稱的對(duì)稱的顯然圖形是關(guān)于顯然圖形是關(guān)于 t當(dāng)n充分(chngfn)大時(shí), 其圖形類似于標(biāo)準(zhǔn)正態(tài)變量概率密度的圖形.,21)(lim22tneth 因?yàn)橐驗(yàn)?)1 , 0(分布分布分布近似于分布近似于足夠大時(shí)足夠大時(shí)所以當(dāng)所以當(dāng)Ntn.)1 , 0(,分分布布相相差差很很大大分分布布與與但但對(duì)對(duì)于于較較小小的的Ntn演 示第10頁/共40頁第十頁，共41頁。t 分布具有分布具有(jyu)下列性質(zhì)：下列性質(zhì)：性質(zhì)5.6 設(shè) , 則當(dāng) 時(shí)有性質(zhì)5.7 設(shè) ， 是T的分布密度，則此

6、性質(zhì)說明(shumng)，當(dāng) 時(shí)，T分布的極限分布是標(biāo)準(zhǔn)正態(tài)分布。)(ntT2n0)(TE2)(nnTD)(ntT)(tp2221)(limtnetpn第11頁/共40頁第十一頁，共41頁。.,),(),(2222的概率分布試求相互獨(dú)立且設(shè)例nYXTYXnYNX得得由定理由定理獨(dú)立獨(dú)立與與則則獨(dú)立獨(dú)立且且又又所以所以因?yàn)橐驗(yàn)榻饨? . 5,),()1 , 0(),(2222 YXYXnYNXNX )(/ )/(/ )(2ntnYXnYXT 第12頁/共40頁第十二頁，共41頁。).,(,),(/,),(),(8 . 52121212212nnFFFnnnYnXFYXnYnX記為記為分布分布的的

7、服從自由度為服從自由度為則稱隨機(jī)變量則稱隨機(jī)變量獨(dú)立獨(dú)立且且設(shè)設(shè)定義定義 分布分布F3.其概率密度為: 其它其它, 00,1222)(2212112221212111ynynnnynnnnynnnn 第13頁/共40頁第十三頁，共41頁。圖圖分布的概率密度曲線如分布的概率密度曲線如F分布有以下性質(zhì)分布有以下性質(zhì)F).,(1),(1221nnFFnnFF則則若若(1) 4(,) 4() 2() 2(2)(),2(,2)(222212122222 nnnnnnnFDnnnFE(2)演 示第14頁/共40頁第十四頁，共41頁。這說明F分布(fnb)極限分布(fnb)也是正態(tài)分布(fnb).dtexF

8、DFEFPxnnnFFtxn22121)()(lim,4),(221 有有對(duì)對(duì)任任意意時(shí)時(shí)則則當(dāng)當(dāng)設(shè)設(shè)(3)第15頁/共40頁第十五頁，共41頁。)1 ,(1), 1(8 . 5222nFFTnFnYXT分布的性質(zhì)知分布的性質(zhì)知由由有有由定義由定義 ,),1(,),(),1 , 0(7 . 5),(2222獨(dú)立獨(dú)立與與且且那么那么獨(dú)立獨(dú)立且且其中其中有有由定義由定義因?yàn)橐驗(yàn)樽C明證明YXXYXnYNXnYXTntT )., 1(),(32nFTntT試證試證已知已知例例第16頁/共40頁第十六頁，共41頁。二、概率分布的分位數(shù).,),10(9 . 5分位數(shù)分位數(shù)的分布的上側(cè)的分布的上側(cè)為為則稱則

9、稱使使若存在若存在和給定的和給定的對(duì)于總體對(duì)于總體定義定義 XxxXPxX u正正態(tài)態(tài)分分布布的的上上側(cè)側(cè)分分位位數(shù)數(shù). 1xeuXPuNNXuxd21101022 ),(),(滿足分位點(diǎn)的上服從標(biāo)準(zhǔn)正態(tài)分布設(shè)第17頁/共40頁第十七頁，共41頁。05. 0u附表附表2-12-1025. 0u根據(jù)(gnj)正態(tài)分布的對(duì)稱性知. uu1,645. 1 ,96. 1 附表附表2-22-2 1)(u 即.,的值可查得由附表給定 u2 )(uuXP 11u第18頁/共40頁第十八頁，共41頁。).()()()(, 10,或分位點(diǎn)或分位點(diǎn)分位數(shù)分位數(shù)分布的上分布的上為為的點(diǎn)的點(diǎn)稱滿足條件稱滿足條件對(duì)于給

10、定的對(duì)于給定的 ntntnttP.分位數(shù)的值分位數(shù)的值得上得上可以通過查表求可以通過查表求 由分布(fnb)的對(duì)稱性知).()(1ntnt .)(, untn時(shí)當(dāng)45)(. 2ntt 分分布布的的上上側(cè)側(cè)分分位位數(shù)數(shù)第19頁/共40頁第十九頁，共41頁。)10(05. 0t附表附表3-13-1,8125. 1 )15(025. 0t.1315. 2 附表附表3-23-2第20頁/共40頁第二十頁，共41頁。.)()()(, 10,2222分分位位數(shù)數(shù)（分分位位點(diǎn)點(diǎn)）分分布布的的上上為為的的點(diǎn)點(diǎn)稱稱滿滿足足條條件件對(duì)對(duì)于于給給定定的的正正數(shù)數(shù) nnnP.,分位點(diǎn)的值分位點(diǎn)的值得上得上可以通過查表

11、求可以通過查表求對(duì)于不同的對(duì)于不同的 n)(. 322n 分分布布的的上上側(cè)側(cè)分分位位數(shù)數(shù)第21頁/共40頁第二十一頁，共41頁。)8(2025. 0 )10(2975. 0 )25(21 . 0 附表附表4-14-1附表4只詳列到 n=45 為止(wizh).,535.17 ,247. 3 附表附表4-24-2.382.34 附表附表4-34-3在在Matlab中求解中求解(qi ji)第22頁/共40頁第二十二頁，共41頁。.2)(,2分位數(shù)分位數(shù)是標(biāo)準(zhǔn)正態(tài)分布的上是標(biāo)準(zhǔn)正態(tài)分布的上其中其中充分大時(shí)充分大時(shí)當(dāng)當(dāng) uunnnn例如例如(lr)6412401201202120120050205

12、0.)(.u .5145利用(lyng)上公式,費(fèi)歇資料費(fèi)歇資料(zlio).,45 分位點(diǎn)的近似值分位點(diǎn)的近似值上上時(shí)時(shí)可以求得可以求得 n費(fèi)歇費(fèi)歇(R.A.Fisher)證明證明:第23頁/共40頁第二十三頁，共41頁。.,),(21可通過查表完成可通過查表完成的值的值求求nnF )7 , 8(025. 0F)14,30(05. 0F附表附表5-15-1,90. 4 .31. 2 附表附表5-25-2.),(),(),(, 10,212121分位數(shù)分位數(shù)分布的上分布的上為為的點(diǎn)的點(diǎn)稱滿足條件稱滿足條件對(duì)于給定的對(duì)于給定的 nnFnnFnnFFP),(214nnFF 分布的上側(cè)分位數(shù)第24頁

13、/共40頁第二十四頁，共41頁。:分位點(diǎn)具有如下性質(zhì)分位點(diǎn)具有如下性質(zhì)分布的上分布的上 F.),(1),(12211nnFnnF 證明證明(zhngmng),(1 211nnFFP 所以所以 ),(11211nnFFP ),(111211nnFFP ,),(111211 nnFFP ),(21nnFF因?yàn)橐驗(yàn)?),(11 211 nnFFP故故第25頁/共40頁第二十五頁，共41頁。),(1 12nnFF因?yàn)橐驗(yàn)?),(1 12 nnFFP所以所以, ),(),(11221-1nnFnnF 比較后得比較后得.),(1),(12211nnFnnF 即即)9 , 21(59 . 0F例例)12,

14、9(105. 0F 8 . 21 .357. 0 . 分位點(diǎn)分位點(diǎn)的一些上的一些上用來求分布表中未列出用來求分布表中未列出 第26頁/共40頁第二十六頁，共41頁。三、小結(jié)(xioji)1.三大(sn d)統(tǒng)計(jì)分布 . , , 2分布分布分布分布分布分布Ft 的定義(dngy),性質(zhì).2.概率分布的分位數(shù)第27頁/共40頁第二十七頁，共41頁。附表2-12-1標(biāo)準(zhǔn)標(biāo)準(zhǔn)(biozhn)正正態(tài)分布表態(tài)分布表z0.000.010.020.030.040.050.060.070.080.090.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.60.500

15、00.53980.57930.61790.65540.69150.72570.75800.78810.81590.84130.86430.88490.90320.91920.93320.94520.50400.54380.58320.62170.65910.69500.72910.76110.79100.81860.84380.86650.88690.90490.92070.93450.94630.50800.54780.58710.62550.66280.69850.73240.76420.79390.82120.84610.86860.88880.90660.92220.93570.947

16、40.51200.55170.59100.62930.66640.70190.73570.76730.79670.82380.84850.87080.89070.90820.92360.93700.94840.51600.55570.59480.63310.67000.70540.73890.77030.79950.82640.85080.87290.89250.90990.92510.93820.94950.51990.55960.59870.63680.67360.70880.74220.77340.80230.82890.85310.87490.89440.91150.92650.939

17、40.95050.52390.56360.60260.64060.67720.71230.74540.77640.80510.83150.85540.87700.89620.91310.92780.94060.95150.52790.56750.60640.64430.68080.71570.74860.77940.80780.83400.85770.87900.89800.91470.92920.94180.95250.53190.57140.61030.64800.68440.71900.75170.78230.81060.83650.85990.88100.89970.91620.930

18、60.94300.95350.53590.57530.61410.65170.68790.72240.75490.78520.81330.83890.86210.88300.90150.91770.93190.94410.95451.645,d21)(22 zttez 第28頁/共40頁第二十八頁，共41頁。附表2-22-2 標(biāo)準(zhǔn)標(biāo)準(zhǔn)(biozhn)正正態(tài)分布表態(tài)分布表z0.000.010.020.030.040.050.060.070.080.091.61.71.81.92.02.12.22.32.42.52.62.72.82.93.00.94520.95540.96410.97130.97

19、720.98210.98610.98930.99180.99380.99530.99650.99740.99810.99870.94630.95640.96480.97190.97780.98260.98640.98960.99200.99400.99550.99660.99750.99820.99900.94740.95730.96560.97260.97830.98300.98680.98980.99220.99410.99560.99670.99760.99820.99930.94840.95820.96640.97320.97880.98340.98710.99010.99250.99

20、430.99570.99680.99770.99830.99950.94950.95910.96710.97380.97930.98380.98710.99040.99270.99450.99590.99690.99770.99840.99970.95050.95990.96780.97440.97980.98420.98780.99060.99290.99460.99600.99700.99780.99840.96980.95150.96080.96860.97500.98030.98460.98810.99090.99310.99480.99610.99710.99790.99850.99

21、980.95250.96160.96930.97560.98080.98500.98840.99110.99320.99490.99620.99720.99790.99850.99990.95350.96250.97000.97620.98120.98540.98870.99130.99340.99510.99630.99730.99800.99860.99990.95450.96330.97060.97670.98170.98530.98900.99160.99360.99520.99640.99740.99810.99861.00001.96,d21)(22 zttez 第29頁/共40頁

22、第二十九頁，共41頁。附表4-14-1= 0.250.100.050.0250.010.005123456789101112131415161.3232.7734.1085.3856.6267.8419.03710.21911.38912.54913.70114.84515.98417.11718.24519.3692.7064.6056.2517.7799.23610.64512.01713.36214.68415.98717.27518.54919.81220.06422.30723.5423.8415.9917.8159.48811.07112.59214.06715.50716.919

23、18.30719.67521.02622.36223.68524.99626.2965.0247.3789.34811.14312.83314.44916.01317.53519.02320.48321.92023.33724.73626.11927.48828.8456.6359.21011.34513.27715.08616.81218.47520.09021.66623.20924.72526.21727.68829.14130.57832.0007.87910.59712.83814.86016.75018.54820.27821.95523.58925.18826.75728.299

24、29.89131.31932.80134.267n2 分布分布(fnb)表表17.535第30頁/共40頁第三十頁，共41頁。=0.9950.990.9750.950.900.75123456789101112131415160.0100.0720.2070.4120.6760.9891.3441.7352.1562.6033.0743.5654.0754.6015.1420.0200.1150.2970.5540.8721.2391.6462.0882.5583.0533.5714.1074.6605.2295.8120.0010.0510.2160.4840.8311.2371.6902.

25、1802.7003.2473.8164.4045.0095.6296.2626.9080.0040.1030.3520.7111.1451.6352.1672.7333.3253.9404.5755.2265.8926.5717.2617.9620.0160.2110.5841.0641.6102.2042.8333.4904.1684.8655.5786.3047.0427.7908.5479.3120.1020.5751.2131.9232.6753.4554.2555.0715.8996.7377.5848.4389.29910.16511.03711.912n3.247附表4-4-2

26、22 分布分布(fnb)表表第31頁/共40頁第三十一頁，共41頁。=0.250.100.050.0250.010.0051718192021222324252627282930313220.48921.60522.71823.82824.93526.03927.14128.24129.33930.43531.52832.62033.71134.80035.88736.97324.76925.98927.20428.41229.61530.81332.00733.19634.38235.56336.74137.91639.08740.25641.42242.58527.58728.86930.

27、14431.41032.67133.92435.17236.41537.65238.88540.11341.33742.55743.77344.98546.19430.19131.52632.85234.17035.47936.78138.07639.36440.64641.92343.19444.46145.71246.97948.23249.48033.40934.80536.19137.56638.93240.28941.63842.98044.31445.64246.96348.27849.58850.89252.19153.48635.71837.15638.58239.99741.

28、40142.79644.18145.55946.92848.29049.64550.99352.33653.67255.00356.328n34.382附表4-4-3 32 分布分布(fnb)表表第32頁/共40頁第三十二頁，共41頁。附表3-13-1 =0.250.100.050.0250.010.005123456789101112131415161.00000.81650.76490.74070.72670.71760.71110.70640.70270.69980.69740.69550.69380.69240.69120.69013.07771.88561.63771.53321.4

29、7591.43981.41491.39681.38301.37221.36341.35621.35021.34501.34061.33686.31382.92002.35342.13182.01501.94321.89461.85951.83311.81251.79591.78231.77091.76131.75311.745912.7062 4.3027 3.1824 2.7764 2.5706 2.4469 2.3646 2.3060 2.2622 2.2281 2.2010 2.1788 2.1604 2.1448 2.1315 2.119931.8207 6.9646 4.5407 3

30、.7469 3.3649 3.1427 2.9980 2.8965 2.8214 2.7638 2.7181 2.6810 2.6503 2.6245 2.6025 2.583563.6574 9.9248 5.8409 4.6041 4.0322 3.7074 3.4995 3.3554 3.2498 3.1693 3.1058 3.0545 3.0123 2.9768 2.9467 2.9208nt分布分布(fnb)表表1.8125第33頁/共40頁第三十三頁，共41頁。附表3-23-2 =0.250.100.050.0250.010.005123456789101112131415161

31、.00000.81650.76490.74070.72670.71760.71110.70640.70270.69980.69740.69550.69380.69240.69120.69013.07771.88561.63771.53321.47591.43981.41491.39681.38301.37221.36341.35621.35021.34501.34061.33686.31382.92002.35342.13182.01501.94321.89461.85951.83311.81251.79591.78231.77091.76131.75311.745912.7062 4.302

32、7 3.1824 2.7764 2.5706 2.4469 2.3646 2.3060 2.2622 2.2281 2.2010 2.1788 2.1604 2.1448 2.1315 2.119931.8207 6.9646 4.5407 3.7469 3.3649 3.1427 2.9980 2.8965 2.8214 2.7638 2.7181 2.6810 2.6503 2.6245 2.6025 2.583563.6574 9.9248 5.8409 4.6041 4.0322 3.7074 3.4995 3.3554 3.2498 3.1693 3.1058 3.0545 3.01

33、23 2.9768 2.9467 2.9208n2.1315t分布分布(fnb)表表第34頁/共40頁第三十四頁，共41頁。附表5-15-1F分布分布(fnb)表表025. 0 1234567891012152024304012012345678910111213141516171819647.838.5117.4412.2210.01 8.81 8.07 7.57 7.21 6.94 6.72 6.55 6.41 6.30 6.20 6.12 6.04 5.95 5.92799.539.0016.0410.65 8.43 7.26 6.54 6.06 5.71 5.46 5.26 5.10

34、4.97 4.86 4.77 4.69 4.62 4.56 4.51864.239.1715.44 9.98 7.76 6.60 5.89 5.42 5.08 4.83 4.63 4.47 4.35 4.24 4.15 4.08 4.01 3.95 3.90899.639.2515.10 9.60 7.39 6.23 5.52 5.50 4.72 4.47 4.28 4.12 4.00 3.89 3.80 3.73 3.66 3.61 3.56921.839.3014.88 9.36 7.15 5.99 5.29 4.82 4.48 4.24 4.04 3.89 3.77 3.66 3.58

35、3.50 3.44 3.38 3.33937.139.3314.73 9.20 6.98 5.82 5.12 4.65 4.23 4.07 3.88 3.73 3.60 3.50 3.41 3.34 3.28 3.22 3.17948.239.3614.62 9.07 6.85 5.70 4.99 4.53 4.20 3.95 3.76 3.61 3.48 3.38 3.29 3.22 3.16 3.10 3.05956.739.3714.54 8.98 6.76 5.60 4.90 4.43 4.10 3.85 3.66 3.51 3.39 3.29 3.20 3.12 3.06 3.01

36、2.96963.339.3914.47 8.90 6.68 5.52 4.82 4.36 4.03 3.78 3.59 3.44 3.31 3.21 3.12 3.05 2.98 2.93 2.88968.639.4014.42 8.84 6.62 5.46 4.76 4.30 3.96 3.72 3.53 3.37 3.25 3.15 3.06 2.99 2.92 2.87 2.82976.739.4114.34 8.75 6.52 5.37 4.67 4.20 3.87 3.62 3.43 3.28 3.15 3.05 2.96 2.89 2.82 2.77 2.72984.939.431

37、4.25 8.66 6.43 5.27 4.57 4.10 3.77 3.52 3.33 3.18 3.05 2.95 2.86 2.79 2.72 2.67 2.62993.139.4514.17 8.56 6.33 5.17 4.47 4.00 3.67 3.42 3.23 3.07 2.95 2.84 2.76 2.68 2.62 2.56 2.51997.239.4614.12 8.51 6.28 5.12 4.42 3.59 3.61 3.37 3.17 3.02 2.89 2.79 2.70 2.63 2.56 2.50 2.45100139.4614.08 8.46 6.23 5

38、.07 4.36 3.89 3.56 3.31 3.12 2.96 2.84 2.73 2.64 2.57 2.50 2.44 2.39100639.4714.04 8.41 6.18 5.01 4.31 3.84 3.51 3.26 3.06 2.91 2.78 2.67 2.59 2.51 2.44 2.38 2.33101439.4913.95 8.31 6.07 4.90 4.20 3.73 3.39 3.14 2.94 2.79 2.66 2.55 2.46 2.38 2.32 2.26 2.20101839.5013.90 8.26 6.02 4.85 4.14 3.67 3.33

39、 3.08 2.88 2.72 2.60 2.49 2.40 2.32 2.25 2.19 2.132n1n4.90第35頁/共40頁第三十五頁，共41頁。 1234567891015202430406012012345678910111213141516171819161.418.5110.13 7.71 6.61 5.99 5.59 5.32 3.12 4.96 4.84 4.75 4.67 4.60 4.54 4.49 4.45 4.41 4.38199.519.00 9.55 6.94 5.79 5.14 4.74 4.46 4.26 4.10 3.98 3.89 3.81 3.74

40、3.68 3.63 3.59 5.55 3.52215.719.16 9.28 6.59 5.41 4.76 4.35 4.07 3.81 3.71 3.59 3.49 3.41 3.34 3.29 3.24 3.20 3.16 3.13224.619.25 9.12 6.39 5.19 4.53 4.12 3.84 3.63 3.48 3.36 3.26 3.18 3.11 3.06 3.01 2.96 2.93 2.90230.219.30 9.01 6.26 5.05 4.39 3.97 3.69 3.48 3.33 3.20 3.11 3.03 2.96 2.90 2.85 2.81

41、2.77 2.74234.019.33 8.94 6.16 4.95 4.28 3.87 3.58 3.37 3.22 3.09 3.00 2.92 2.85 2.79 2.74 2.70 2.66 2.63236.819.35 8.89 6.09 4.88 4.21 3.79 3.50 3.29 3.14 3.01 2.91 2.83 2.76 2.71 2.66 2.61 2.58 2.54238.919.37 8.85 6.04 4.82 4.15 3.73 3.44 3.23 3.07 2.95 2.85 2.77 2.70 2.64 2.59 2.55 2.51 2.4824.0519.38 8.81 6.00 4.77 4.10 3.68 3.39 3.18 3.02 2.90 2.80 2.71 2.65 2.59 2.54 2.49 2.46 2.42241.919.40 8.79 5.96 4.74 4.06 3.64 3.35 3.14 2.98 2.85 2.75 2.67 2.60 2.54 2.49 2.45 2.41 2.38245.919.43 8.70 5.86 4.62 3.94 2.51 3.22 3.01 2.85 2.72 2.62