常用統(tǒng)計分布

1、5.2 常用統(tǒng)計分布一、常見分布一、常見分布二、概率分布的分位數(shù)二、概率分布的分位數(shù)三、小結(jié)三、小結(jié)一、常見分布122222122225.6,(0, 1),( ).nnnnNnn定 義 設(shè)相 互 獨(dú) 立 ， 同 服 從分 布 則 稱 統(tǒng) 計 量服 從 自 由度 為的分 布 記 為222212:.nn自由度 指中右端包含獨(dú)立變量的個數(shù)分布分布2 1.隨機(jī)數(shù)隨機(jī)數(shù)演示演示分布函數(shù)與密度函數(shù)演示分布函數(shù)與密度函數(shù)演示分布的概率密度為分布的概率密度為定理定理)(4 . 52n 其其它它00)2(21)(2122xexnxpxnn證明證明,21,21)1(2分分布布分分布布即即為為因因為為 (0,1),

2、iN又因為22(1),i由定義21 1,1, 2, .2 2iin即.)(2圖圖分布的概率密度曲線如分布的概率密度曲線如n 12,n 因為相互獨(dú)立22212,n所以也相互獨(dú)立分布的可加性知分布的可加性知根據(jù)根據(jù) 221nnii.21,2 n分布的性質(zhì)分布的性質(zhì)2 性質(zhì)性質(zhì)1).(,),(),(2122221222122221221nnnn 則則立立獨(dú)獨(dú)并并且且設(shè)設(shè))(2分布的可加性分布的可加性 (此性質(zhì)可以推廣到多個隨機(jī)變量的情形此性質(zhì)可以推廣到多個隨機(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)性質(zhì)性質(zhì)2.2)(,)()

3、,(2222nDnEn 則則若若證明證明(0,1),iN因為21,iiED所以2422()() ()iiiDEE, 213 ., 2, 1ni 221 ()niiEE故21()niiE,n 221()niiDD21()niiD.2n )(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 性質(zhì)性質(zhì)3dtexnnPxntxnnn22212lim,),(222 有有則對任意則對任意設(shè)設(shè)2212122212,(0,1),nniniinXN 證明 由假設(shè)其中獨(dú)立且每個因而獨(dú)立同分布且22()1,()2(1,2, )iiEDin2lim2xnnPnn 由中心極限定理得由中心極限定理得 xniindtexnnXP

4、t2221lim12).2 ,().1 , 0(2,222nnNNnnnnn近似近似進(jìn)而進(jìn)而近似服從近似服從很大時很大時當(dāng)當(dāng)也即也即分布分布分布的極限分布是正態(tài)分布的極限分布是正態(tài)即即 12612221122345621,(0,1),()().NC CYCC 例設(shè)為來自正態(tài)總體的一組樣本 求使得服從分布1212(0,2),(0,1)2NN解則34563456(0,4),(0,1)4NN則同理同理122且34564X與與相互獨(dú)立相互獨(dú)立212()2所以223456() (2)4.,412121CC則則25.7(0,1),( ),/, ( ).NnTnntTt n 定義設(shè)且獨(dú)立 則稱隨機(jī)變量服從自

5、由度為的 分布 記為t 分布又稱分布又稱學(xué)生氏學(xué)生氏(Student)分布分布.學(xué)生氏資料學(xué)生氏資料 tntnnnthn,1221)(212 分布的概率密度函數(shù)為分布的概率密度函數(shù)為)(nt分布分布t2.圖圖分布的概率密度曲線如分布的概率密度曲線如t.0對稱的對稱的顯然圖形是關(guān)于顯然圖形是關(guān)于 t當(dāng)當(dāng)n充分大時充分大時, 其圖其圖形類似于標(biāo)準(zhǔn)正態(tài)形類似于標(biāo)準(zhǔn)正態(tài)變量概率密度的圖變量概率密度的圖形形.,21)(lim22tneth 因為因為,)1 , 0(分分布布分分布布近近似似于于足足夠夠大大時時所所以以當(dāng)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對于于較較小小的的N

6、tnt 分布具有下列性質(zhì)：性質(zhì)性質(zhì)5.6 設(shè)設(shè) , 則當(dāng)則當(dāng) 時有時有性質(zhì)性質(zhì)5.7 設(shè)設(shè) ， 是是T的分布密度，的分布密度，則則此性質(zhì)說明，當(dāng)此性質(zhì)說明，當(dāng) 時，時，T分布的極限分布的極限分布是標(biāo)準(zhǔn)正態(tài)分布。分布是標(biāo)準(zhǔn)正態(tài)分布。)(ntT2n0)(TE2)(nnTD)(ntT)(tp2221)(limtnetpn2222( ,),( ),.NnTn 例設(shè)且相互獨(dú)立 試求的概率分布2222( ,),(0,1)( ),NNnX Y 解因為所以又且獨(dú)立 則與獨(dú)立由定理得2()/ ( )( /)/Tt nnn 22121122125.8(),(),/(,),/(,).nnnFnnFnFF nn 定義

7、設(shè)且獨(dú)立 則稱隨機(jī)變量服從自由度的分布記為分分布布F3.分分布布的的概概率率密密度度為為),(21nnF 其它其它, 00,1222)(2212112221212111ynynnnynnnnynnnn 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)(,)()()()(),(,)(4422222222212122222nnnnnnnFDnnnFE(2)這說明這說明F分布極限分布也是正態(tài)分布分布極限分布也是正態(tài)分布.dtexFDFEFPxnnnFFtxn22121)()(lim,4),(221有有對對任任意意時時則

8、則當(dāng)當(dāng)設(shè)設(shè)(3)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 2222 ( ),5.7(0,1),( ),(1),Tt nXTY nNn 證明 因為由定義有其中且獨(dú)立 那么且與 獨(dú)立.)., 1(),(32nFTntT試證試證已知已知例例二、概率分布的分位數(shù)5.9(01),.xPxx定義對于總體 和給定的若存在使則稱 為 的分布的上側(cè) 分位數(shù) 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). 122(0,1),(0,1)1d2xuNNuPuex設(shè)服從標(biāo)準(zhǔn)正態(tài)分布的上分位點滿足05. 0u025. 0u根據(jù)正態(tài)分布的對稱性知根據(jù)正態(tài)分布的對稱性

9、知. uu1,645. 1 ,96. 1 1)(u 即.,的值可查得由附表給定 u211()Puu ).()()()(, 10,或分位點或分位點分位數(shù)分位數(shù)分布的上分布的上為為的點的點稱滿足條件稱滿足條件對于給定的對于給定的 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 由分布的對稱性知由分布的對稱性知).()(1ntnt .)(, untn時當(dāng)45)(. 2ntt 分分布布的的上上側(cè)側(cè)分分位位數(shù)數(shù))10(05. 0t,8125. 1 )15(025. 0t.1315. 2 .)()()(, 10,2222分分位位數(shù)數(shù)（分分位位點點）分分布布的的上上為為的的點點稱

10、稱滿滿足足條條件件對對于于給給定定的的正正數(shù)數(shù) nnnP.,分位點的值分位點的值得上得上可以通過查表求可以通過查表求對于不同的對于不同的 n)(. 322n 分分布布的的上上側(cè)側(cè)分分位位數(shù)數(shù))8(2025. 0 )10(2975. 0 )25(21 . 0 附表附表4只詳列到只詳列到 n=45 為止為止.,535.17 ,247. 3 .382.34 .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充分大時當(dāng)當(dāng) uunnnn例如例如64124012012021201200502050.)(.u .5145利用上公式利用上公式,費(fèi)歇資料費(fèi)歇資料而查詳表可得而查詳表可得

11、.505.67)50(205. 0 .,45 分分位位點點的的近近似似值值上上時時可可以以求求得得 n費(fèi)歇費(fèi)歇(R.A.Fisher)證明證明:.,),(21可可通通過過查查表表完完成成的的值值求求nnF )7 , 8(025. 0F)14,30(05. 0F,90. 4 .31. 2 .),(),(),(, 10,212121分分位位數(shù)數(shù)分分布布的的上上為為的的點點稱稱滿滿足足條條件件對對于于給給定定的的 nnFnnFnnFFP),(214nnFF 分布的上側(cè)分位數(shù):分位點具有如下性質(zhì)分位點具有如下性質(zhì)分布的上分布的上 F.),(1),(12211nnFnnF 證明證明),(1 211nnF

12、FP 所所以以 ),(11211nnFFP ),(111211nnFFP ,),(111211 nnFFP ),(21nnFF因因為為,),(11 211 nnFFP故故),(1 12nnFF因為因為,),(1 12 nnFFP所所以以, ),(),(11221-1nnFnnF 比比較較后后得得.),(1),(12211nnFnnF 即即)9 , 21(59 . 0F例例)12, 9(105. 0F 8 . 21 .357. 0 . 分分位位點點的的一一些些上上用用來來求求分分布布表表中中未未列列出出 三、小結(jié)1.三個來自正態(tài)分布的抽樣分布三個來自正態(tài)分布的抽樣分布: . , , 2分分布布分

13、分布布分分布布Ft 的定義的定義,性質(zhì)性質(zhì).2.分位數(shù)的概念分位數(shù)的概念.辛欽定理), 2 , 1()(,21 kXEXXXkn 望望一一分分布布，且且具具有有數(shù)數(shù)學(xué)學(xué)期期相相互互獨(dú)獨(dú)立立，服服從從同同設(shè)設(shè)隨隨機(jī)機(jī)變變量量. 11lim,1 nkknXnP有有則則對對于于任任意意正正數(shù)數(shù)附表2-1標(biāo)準(zhǔn)正態(tài)分布表標(biāo)準(zhǔn)正態(tài)分布表z0.000.010.020.030.040.050.060.070.080.090.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.60.50000.53980.57930.61790.65540.69150.72570.7

14、5800.78810.81590.84130.86430.88490.90320.91920.93320.94520.50400.54380.58320.62170.65910.69500.72910.76110.79100.81860.84380.86650.88690.90490.92070.93450.94630.50800.54780.58710.62550.66280.69850.73240.76420.79390.82120.84610.86860.88880.90660.92220.93570.94740.51200.55170.59100.62930.66640.70190.7

15、3570.76730.79670.82380.84850.87080.89070.90820.92360.93700.94840.51600.55570.59480.63310.67000.70540.73890.77030.79950.82640.85080.87290.89250.90990.92510.93820.94950.51990.55960.59870.63680.67360.70880.74220.77340.80230.82890.85310.87490.89440.91150.92650.93940.95050.52390.56360.60260.64060.67720.7

16、1230.74540.77640.80510.83150.85540.87700.89620.91310.92780.94060.95150.52790.56750.60640.64430.68080.71570.74860.77940.80780.83400.85770.87900.89800.91470.92920.94180.95250.53190.57140.61030.64800.68440.71900.75170.78230.81060.83650.85990.88100.89970.91620.93060.94300.95350.53590.57530.61410.65170.6

17、8790.72240.75490.78520.81330.83890.86210.88300.90150.91770.93190.94410.95451.645附表2-2標(biāo)準(zhǔn)正態(tài)分布表標(biāo)準(zhǔn)正態(tài)分布表z0.000.010.020.030.040.050.060.070.080.091.61.71.81.92.02.12.22.32.42.52.62.72.82.93.00.94520.95540.96410.97130.97720.98210.98610.98930.99180.99380.99530.99650.99740.99810.99870.94630.95640.96480.97190

18、.97780.98260.98640.98960.99200.99400.99550.99660.99750.99820.99900.94740.95730.96560.97260.97830.98300.98680.98980.99220.99410.99560.99670.99760.99820.99930.94840.95820.96640.97320.97880.98340.98710.99010.99250.99430.99570.99680.99770.99830.99950.94950.95910.96710.97380.97930.98380.98710.99040.99270

19、.99450.99590.99690.99770.99840.99970.95050.95990.96780.97440.97980.98420.98780.99060.99290.99460.99600.99700.99780.99840.96980.95150.96080.96860.97500.98030.98460.98810.99090.99310.99480.99610.99710.99790.99850.99980.95250.96160.96930.97560.98080.98500.98840.99110.99320.99490.99620.99720.99790.99850

20、.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附表4-1= 0.250.100.050.0250.010.005123456789101112131415161.3232.7734.1085.3856.6267.8419.03710.21911.38912.54913

21、.70114.84515.98417.11718.24519.3692.7064.6056.2517.7799.23610.64512.01713.36214.68415.98717.27518.54919.81220.06422.30723.5423.8415.9917.8159.48811.07112.59214.06715.50716.91918.30719.67521.02622.36223.68524.99626.2965.0247.3789.34811.14312.83314.44916.01317.53519.02320.48321.92023.33724.73626.11927

22、.48828.8456.6359.21011.34513.27715.08616.81218.47520.09021.66623.20924.72526.21727.68829.14130.57832.0007.87910.59712.83814.86016.75018.54820.27821.95523.58925.18826.75728.29929.89131.31932.80134.267n2 分布表分布表17.535=0.9950.990.9750.950.900.75123456789101112131415160.0100.0720.2070.4120.6760.9891.3441

23、.7352.1562.6033.0743.5654.0754.6015.1420.0200.1150.2970.5540.8721.2391.6462.0882.5583.0533.5714.1074.6605.2295.8120.0010.0510.2160.4840.8311.2371.6902.1802.7003.2473.8164.4045.0095.6296.2626.9080.0040.1030.3520.7111.1451.6352.1672.7333.3253.9404.5755.2265.8926.5717.2617.9620.0160.2110.5841.0641.6102

24、.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4-22 分布表分布表=0.250.100.050.0250.010.0051718192021222324252627282930313220.48921.60522.71823.82824.93526.03927.14128.24129.33930.43531.52832.62033.71134.8003

25、5.88736.97324.76925.98927.20428.41229.61530.81332.00733.19634.38235.56336.74137.91639.08740.25641.42242.58527.58728.86930.14431.41032.67133.92435.17236.41537.65238.88540.11341.33742.55743.77344.98546.19430.19131.52632.85234.17035.47936.78138.07639.36440.64641.92343.19444.46145.71246.97948.23249.4803

26、3.40934.80536.19137.56638.93240.28941.63842.98044.31445.64246.96348.27849.58850.89252.19153.48635.71837.15638.58239.99741.40142.79644.18145.55946.92848.29049.64550.99352.33653.67255.00356.328n34.382附表4-32 分布表分布表附表3-1 =0.250.100.050.0250.010.005123456789101112131415161.00000.81650.76490.74070.72670.7

27、1760.71110.70640.70270.69980.69740.69550.69380.69240.69120.69013.07771.88561.63771.53321.47591.43981.41491.39681.38301.37221.36341.35621.35021.34501.34061.33686.31382.92002.35342.13182.01501.94321.89461.85951.83311.81251.79591.78231.77091.76131.75311.745912.7062 4.3027 3.1824 2.7764 2.5706 2.4469 2.

28、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 3.0123 2.9768 2.9467 2.9208nt分布表分布表1

29、.8125附表3-2 =0.250.100.050.0250.010.005123456789101112131415161.00000.81650.76490.74070.72670.71760.71110.70640.70270.69980.69740.69550.69380.69240.69120.69013.07771.88561.63771.53321.47591.43981.41491.39681.38301.37221.36341.35621.35021.34501.34061.33686.31382.92002.35342.13182.01501.94321.89461.859

30、51.83311.81251.79591.78231.77091.76131.75311.745912.7062 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 3.7469 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 4.604

31、1 4.0322 3.7074 3.4995 3.3554 3.2498 3.1693 3.1058 3.0545 3.0123 2.9768 2.9467 2.9208n2.1315t分布表分布表附表5-1F分布表分布表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

32、.54 6.06 5.71 5.46 5.26 5.10 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

33、.24 4.04 3.89 3.77 3.66 3.58 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

34、.39 3.29 3.20 3.12 3.06 3.01 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

35、.89 2.82 2.77 2.72984.939.4314.25 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

36、.45100139.4614.08 8.46 6.23 5.07 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

37、8.26 6.02 4.85 4.14 3.67 3.33 3.08 2.88 2.72 2.60 2.49 2.40 2.32 2.25 2.19 2.132n1n4.90 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

38、.81 3.74 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

39、.85 2.81 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 2.4824.0519.3