常用统计分布

1、常用统计分布常用统计分布一、常见分布一、常见分布二、概率分布的分位数二、概率分布的分位数三、小结三、小结一、常见分布一、常见分布).(,) 1, 0(,6 . 522222221221nnXXXNXXXnn记为分布的度为服从自由则称统计量分布相互独立，同服从、设定义.:222212变量的个数中右端包含独立指自由度nXXX分布（卡方分布）分布（卡方分布）2. 1 分布的概率密度为分布的概率密度为定理定理)(4 . 52n 其其它它00)2(21)(2122xexnxpxnn.)(2图图分布的概率密度曲线如分布的概率密度曲线如n 分布的性质分布的性质2 性质性质1).(,),(),(2122221

2、222122221221nnnn 则则立立独独并并且且设设)(2分布的可加性分布的可加性 (此性质可以推广到多个随机变量的情形此性质可以推广到多个随机变量的情形).(,), 2, 1(),(21212222mmiiiiinnnmin 则则独独立立相相互互并并且且设设性质性质2.2)(,)(),(2222nDnEn 则则若若证明证明),1, 0( NXi因因为为, 1)()(2 iiXDXE所以所以2242)()()(iiiXEXEXD )(2分布的数学期望和方差分布的数学期望和方差 dxexXExi24422123221xedxdxexexxx22232221321 niiXEE122)( 故

3、故 niiXDD122)( niiXE12)(,n niiXD12)(.2n )(2iXD, 213 ., 2, 1ni 22213xedx3dxexexx22222133性质性质3dtexnnPxntxnn22212lim,),(222有则对任意设且独立同分布因而独立且每个其中由假设和定义证明,),1 , 0(, 6 . 52222121122ninniiXXXNXXXXX), 2 , 1(2)(, 1)(22niXDXEii 2lim2xnnPn由中心极限定理得xniindtexnnXPt2221lim12).2 ,().1 , 0(2,222nnNNnnn近似进而近似服从很大时当也即分布

4、分布的极限分布是正态即.)()(,)1 , 0(,12265432221121621分分布布服服从从使使得得求求样样本本的的一一组组为为来来自自正正态态总总体体设设例例 XXXXCXXCYCCNXXX )1 , 0(2),2 , 0(2121NXXNXX 则则解解),(),(1044065436543NXXXXNXXXX则则同理同理221XX 且且46543XXXX与与相互独立相互独立2212)(XX 所所以以)2()4(226543 XXXX .41,2121 CC则则).(,/,),(),(.ntTtnnYXTYXnYNX记为分布的服从自由度为则称随机变量独立且设定义21075 tntnn

5、nthn,1221)(212 分布的概率密度函数为分布的概率密度函数为)(nt分布分布t2.图图分布的概率密度曲线如分布的概率密度曲线如t.0对称的对称的显然图形是关于显然图形是关于 t当当n充分大时充分大时, 其图其图形类似于标准正态形类似于标准正态变量概率密度的图变量概率密度的图形形.,21)(lim22tneth 因为因为,)1 , 0(分分布布分分布布近近似似于于足足够够大大时时所所以以当当Ntn.)1 , 0(,分分布布相相差差很很大大分分布布与与但但对对于于较较小小的的Ntn演演 示示t 分布分布具有下列性质：具有下列性质：性质性质5.6 设设 , 则当则当 时有时有性质性质5.7

6、 设设 ， 是是T的分布密度，的分布密度，则则此性质说明，当此性质说明，当 时，时，T分布的极限分布的极限分布是标准正态分布。分布是标准正态分布。)(ntT2n0)(TE2)(nnTD)(ntT)(tp2221)(limtnetpn.,),(),(2222的概率分布试求相互独立且设例nYXTYXnYNX得得由定理由定理独立独立与与则则独立独立且且又又所以所以因为因为解解7 . 5,),()1 , 0(),(2222 YXYXnYNXNX )(/ )/(/ )(2ntnYXnYXT ).,(,),(/,),(),(8 . 52121212212nnFFFnnnYnXFYXnYnX记记为为分分布布

7、的的服服从从自自由由度度为为则则称称随随机机变变量量独独立立且且设设定定义义 分分布布F3.其概率密度为其概率密度为: 其它其它, 00,1222)(2212112221212111ynynnnynnnnynnnn 图图分布的概率密度曲线如分布的概率密度曲线如F分布有以下性质分布有以下性质F).,(1),(1221nnFFnnFF则则若若(1) 4(,) 4() 2() 2(2)(),2(,2)(222212122222 nnnnnnnFDnnnFE(2)演演 示示这说明这说明F分布极限分布也是正态分布分布极限分布也是正态分布.dtexFDFEFPxnnnFFtxn22121)()(lim,4

8、),(221 有有对对任任意意时时则则当当设设(3)1 ,(1), 1(8 . 5222nFFTnFnYXT分布的性质知分布的性质知由由有有由定义由定义 ,),1(,),(),1 , 0(7 . 5),(2222独立独立与与且且那么那么独立独立且且其中其中有有由定义由定义因为因为证明证明YXXYXnYNXnYXTntT )., 1(),(32nFTntT试证试证已知已知例例二、概率分布的分位数二、概率分布的分位数.,),10(9 . 5分位数分位数的分布的上侧的分布的上侧为为则称则称使使若存在若存在和给定的和给定的对于总体对于总体定义定义 XxxXPxX u正正态态分分布布的的上上侧侧分分位位

9、数数. 1xeuXPuNNXuxd21101022 ),(),(满足分位点的上服从标准正态分布设05. 0u附表附表2-12-1025. 0u根据正态分布的对称性知根据正态分布的对称性知. uu1,645. 1 ,96. 1 附表附表2-22-2 1)(u 即.,的值可查得由附表给定 u2 )(uuXP 11u).()()()(, 10,或分位点或分位点分位数分位数分布的上分布的上为为的点的点称满足条件称满足条件对于给定的对于给定的 ntntnttP.分位数的值分位数的值得上得上可以通过查表求可以通过查表求 由分布的对称性知由分布的对称性知).()(1ntnt .)(, untn时当45)(.

10、 2ntt 分分布布的的上上侧侧分分位位数数)10(05. 0t附表附表3-13-1,8125. 1 )15(025. 0t.1315. 2 附表附表3-23-2.)()()(, 10,2222分分位位数数（分分位位点点）分分布布的的上上为为的的点点称称满满足足条条件件对对于于给给定定的的正正数数 nnnP.,分位点的值分位点的值得上得上可以通过查表求可以通过查表求对于不同的对于不同的 n)(. 322n 分分布布的的上上侧侧分分位位数数)8(2025. 0 )10(2975. 0 )25(21 . 0 附表附表4-14-1附表附表4只详列到只详列到 n=45 为止为止.,535.17 ,24

11、7. 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.2)(,2分位数分位数是标准正态分布的上是标准正态分布的上其中其中充分大时充分大时当当 uunnnn例如例如64124012012021201200502050.)(.u .5145利用上公式利用上公式,费歇资料费歇资料.,45 分分位位点点的的近近似似值值上上时时可可以以求求得得 n费歇费歇(R.A.Fisher)证明证明:.,),(21可可通通过过查查表表完完成成的的值值求求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-

12、25-2.),(),(),(, 10,212121分分位位数数分分布布的的上上为为的的点点称称满满足足条条件件对对于于给给定定的的 nnFnnFnnFFP),(214nnFF 分布的上侧分位数:分位点具有如下性质分位点具有如下性质分布的上分布的上 F.),(1),(12211nnFnnF 证明证明),(1 211nnFFP 所所以以 ),(11211nnFFP ),(111211nnFFP ,),(111211 nnFFP ),(21nnFF因因为为,),(11 211 nnFFP故故),(1 12nnFF因为因为,),(1 12 nnFFP所所以以, ),(),(11221-1nnFnnF

13、比比较较后后得得.),(1),(12211nnFnnF 即即)9 , 21(59 . 0F例例)12, 9(105. 0F 8 . 21 .357. 0 . 分分位位点点的的一一些些上上用用来来求求分分布布表表中中未未列列出出 三、小结三、小结1.三大统计分布三大统计分布 . , , 2分分布布分分布布分分布布Ft 的定义的定义,性质性质.2.概率分布的分位数附表附表2-12-1 标准正态分布表标准正态分布表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.

14、50000.53980.57930.61790.65540.69150.72570.75800.78810.81590.84130.86430.88490.90320.91920.93320.94520.50400.54380.58320.62170.65910.69500.72910.76110.79100.81860.84380.86650.88690.90490.92070.93450.94630.50800.54780.58710.62550.66280.69850.73240.76420.79390.82120.84610.86860.88880.90660.92220.93570.

15、94740.51200.55170.59100.62930.66640.70190.73570.76730.79670.82380.84850.87080.89070.90820.92360.93700.94840.51600.55570.59480.63310.67000.70540.73890.77030.79950.82640.85080.87290.89250.90990.92510.93820.94950.51990.55960.59870.63680.67360.70880.74220.77340.80230.82890.85310.87490.89440.91150.92650.

16、93940.95050.52390.56360.60260.64060.67720.71230.74540.77640.80510.83150.85540.87700.89620.91310.92780.94060.95150.52790.56750.60640.64430.68080.71570.74860.77940.80780.83400.85770.87900.89800.91470.92920.94180.95250.53190.57140.61030.64800.68440.71900.75170.78230.81060.83650.85990.88100.89970.91620.

17、93060.94300.95350.53590.57530.61410.65170.68790.72240.75490.78520.81330.83890.86210.88300.90150.91770.93190.94410.95451.645,d21)(22 zttez 附表附表2-22-2标准正态分布表标准正态分布表z0.000.010.020.030.040.050.060.070.080.091.61.71.81.92.02.12.22.32.42.52.62.72.82.93.00.94520.95540.96410.97130.97720.98210.98610.98930.99

18、180.99380.99530.99650.99740.99810.99870.94630.95640.96480.97190.97780.98260.98640.98960.99200.99400.99550.99660.99750.99820.99900.94740.95730.96560.97260.97830.98300.98680.98980.99220.99410.99560.99670.99760.99820.99930.94840.95820.96640.97320.97880.98340.98710.99010.99250.99430.99570.99680.99770.99

19、830.99950.94950.95910.96710.97380.97930.98380.98710.99040.99270.99450.99590.99690.99770.99840.99970.95050.95990.96780.97440.97980.98420.98780.99060.99290.99460.99600.99700.99780.99840.96980.95150.96080.96860.97500.98030.98460.98810.99090.99310.99480.99610.99710.99790.99850.99980.95250.96160.96930.97

20、560.98080.98500.98840.99110.99320.99490.99620.99720.99790.99850.99990.95350.96250.97000.97620.98120.98540.98870.99130.99340.99510.99630.99730.99800.99860.99990.95450.96330.97060.97670.98170.98530.98900.99160.99360.99520.99640.99740.99810.99861.00001.96,d21)(22 zttez 附表附表4-14-1= 0.250.100.050.0250.01

21、0.005123456789101112131415161.3232.7734.1085.3856.6267.8419.03710.21911.38912.54913.70114.84515.98417.11718.24519.3692.7064.6056.2517.7799.23610.64512.01713.36214.68415.98717.27518.54919.81220.06422.30723.5423.8415.9917.8159.48811.07112.59214.06715.50716.91918.30719.67521.02622.36223.68524.99626.296

22、5.0247.3789.34811.14312.83314.44916.01317.53519.02320.48321.92023.33724.73626.11927.48828.8456.6359.21011.34513.27715.08616.81218.47520.09021.66623.20924.72526.21727.68829.14130.57832.0007.87910.59712.83814.86016.75018.54820.27821.95523.58925.18826.75728.29929.89131.31932.80134.267n2 分布表分布表17.535=0.

23、9950.990.9750.950.900.75123456789101112131415160.0100.0720.2070.4120.6760.9891.3441.7352.1562.6033.0743.5654.0754.6015.1420.0200.1150.2970.5540.8721.2391.6462.0882.5583.0533.5714.1074.6605.2295.8120.0010.0510.2160.4840.8311.2371.6902.1802.7003.2473.8164.4045.0095.6296.2626.9080.0040.1030.3520.7111.1

24、451.6352.1672.7333.3253.9404.5755.2265.8926.5717.2617.9620.0160.2110.5841.0641.6102.2042.8333.4904.1684.8655.5786.3047.0427.7908.5479.3120.1020.5751.2131.9232.6753.4554.2555.0715.8996.7377.5848.4389.29910.16511.03711.912n3.247附表附表4-4-2 22 分布表分布表=0.250.100.050.0250.010.0051718192021222324252627282930

25、313220.48921.60522.71823.82824.93526.03927.14128.24129.33930.43531.52832.62033.71134.80035.88736.97324.76925.98927.20428.41229.61530.81332.00733.19634.38235.56336.74137.91639.08740.25641.42242.58527.58728.86930.14431.41032.67133.92435.17236.41537.65238.88540.11341.33742.55743.77344.98546.19430.19131

26、.52632.85234.17035.47936.78138.07639.36440.64641.92343.19444.46145.71246.97948.23249.48033.40934.80536.19137.56638.93240.28941.63842.98044.31445.64246.96348.27849.58850.89252.19153.48635.71837.15638.58239.99741.40142.79644.18145.55946.92848.29049.64550.99352.33653.67255.00356.328n34.382附表附表4-4-3 32

27、分布表分布表附表附表3-13-1 =0.250.100.050.0250.010.005123456789101112131415161.00000.81650.76490.74070.72670.71760.71110.70640.70270.69980.69740.69550.69380.69240.69120.69013.07771.88561.63771.53321.47591.43981.41491.39681.38301.37221.36341.35621.35021.34501.34061.33686.31382.92002.35342.13182.01501.94321.894

28、61.85951.83311.81251.79591.78231.77091.76131.75311.745912.7062 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 3.7469 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

29、 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1.8125附表附表3-23-2 =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.

30、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 3.7469 3.364

31、9 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 2.9208n2.1315t分布表分布表附表附表5-15-1F分布表分布表025. 0 1234567891012152024304012012345678910111213141516171819647.838.5117.4412

32、.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 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

33、.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 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

34、.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 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

35、.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 2.72984.939.4314.25 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

36、.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 5.07 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 2.3

37、3101439.4913.95 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 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

38、.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 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

39、.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 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