常用统计分布

1、5.2 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122222122225.6,(0, 1),( ).nnnnNnn定 义 设相 互 独 立 ， 同 服 从分 布 则 称 统 计 量服 从 自 由度 为的分 布 记 为222212:.nn自由度 指中右端包含独立变量的个数分布分布2 1.随机数随机数演示演示分布函数与密度函数演示分布函数与密度函数演示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证明证明,21,21)1(2分分布布分分布布即即为为因因为为 (0,1),

2、iN又因为22(1),i由定义21 1,1, 2, .2 2iin即.)(2图图分布的概率密度曲线如分布的概率密度曲线如n 12,n 因为相互独立22212,n所以也相互独立分布的可加性知分布的可加性知根据根据 221nnii.21,2 n分布的性质分布的性质2 性质性质1).(,),(),(2122221222122221221nnnn 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 则则独立独立相互相互并且并且设设性质性质2.2)(,)()

3、,(2222nDnEn 则则若若证明证明(0,1),iN因为21,iiED所以2422()() ()iiiDEE, 213 ., 2, 1ni 221 ()niiEE故21()niiE,n 221()niiDD21()niiD.2n )(2分布的数学期望和方差分布的数学期望和方差 性质性质3dtexnnPxntxnnn22212lim,),(222 有有则对任意则对任意设设2212122212,(0,1),nniniinXN 证明 由假设其中独立且每个因而独立同分布且22()1,()2(1,2, )iiEDin2lim2xnnPnn 由中心极限定理得由中心极限定理得 xniindtexnnXP

4、t2221lim12).2 ,().1 , 0(2,222nnNNnnnnn近似近似进而进而近似服从近似服从很大时很大时当当也即也即分布分布分布的极限分布是正态分布的极限分布是正态即即 12612221122345621,(0,1),()().NC CYCC 例设为来自正态总体的一组样本 求使得服从分布1212(0,2),(0,1)2NN解则34563456(0,4),(0,1)4NN则同理同理122且34564X与与相互独立相互独立212()2所以223456() (2)4.,412121CC则则25.7(0,1),( ),/, ( ).NnTnntTt n 定义设且独立 则称随机变量服从自

5、由度为的 分布 记为t 分布又称分布又称学生氏学生氏(Student)分布分布.学生氏资料学生氏资料 tntnnnthn,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N

6、tnt 分布具有下列性质：性质性质5.6 设设 , 则当则当 时有时有性质性质5.7 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)(limtnetpn2222( ,),( ),.NnTn 例设且相互独立 试求的概率分布2222( ,),(0,1)( ),NNnX Y 解因为所以又且独立 则与独立由定理得2()/ ( )( /)/Tt nnn 22121122125.8(),(),/(,),/(,).nnnFnnFnFF nn 定义

7、设且独立 则称随机变量服从自由度的分布记为分分布布F3.分分布布的的概概率率密密度度为为),(21nnF 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)(,)()()()(),(,)(4422222222212122222nnnnnnnFDnnnFE(2)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),(221有有对对任任意意时时则

8、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 2222 ( ),5.7(0,1),( ),(1),Tt nXTY nNn 证明 因为由定义有其中且独立 那么且与 独立.)., 1(),(32nFTntT试证试证已知已知例例二、概率分布的分位数5.9(01),.xPxx定义对于总体 和给定的若存在使则称 为 的分布的上侧 分位数 u正正态态分分布布的的上上侧侧分分位位数数. 122(0,1),(0,1)1d2xuNNuPuex设服从标准正态分布的上分位点满足05. 0u025. 0u根据正态分布的对称性知根据正态分布的对称性

9、知. uu1,645. 1 ,96. 1 1)(u 即.,的值可查得由附表给定 u211()Puu ).()()()(, 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)(. 2ntt 分分布布的的上上侧侧分分位位数数)10(05. 0t,8125. 1 )15(025. 0t.1315. 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

10、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只详列到只详列到 n=45 为止为止.,535.17 ,247. 3 .382.34 .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,费歇资料费歇资料而查详表可得而查详表可得

11、.505.67)50(205. 0 .,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,90. 4 .31. 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 211nnF

12、FP 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 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1.三个来自正态分布的抽样分布三个来自正态分布的抽样分布: . , , 2分分布布分

13、分布布分分布布Ft 的定义的定义,性质性质.2.分位数的概念分位数的概念.辛钦定理), 2 , 1()(,21 kXEXXXkn 望望一一分分布布，且且具具有有数数学学期期相相互互独独立立，服服从从同同设设随随机机变变量量. 11lim,1 nkknXnP有有则则对对于于任任意意正正数数附表2-1标准正态分布表标准正态分布表z0.000.010.020.030.040.050.060.070.080.090.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.60.50000.53980.57930.61790.65540.69150.72570.7

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z0.000.010.020.030.040.050.060.070.080.091.61.71.81.92.02.12.22.32.42.52.62.72.82.93.00.94520.95540.96410.97130.97720.98210.98610.98930.99180.99380.99530.99650.99740.99810.99870.94630.95640.96480.97190

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