統(tǒng)計(jì)量與抽樣分布概述

統(tǒng)計(jì)量與抽樣分布概述總體選擇個(gè)體樣本觀測(cè)樣本樣本觀察值(數(shù)據(jù))數(shù)據(jù)處理樣本有關(guān)結(jié)論推斷總體性質(zhì)

統(tǒng)計(jì)量統(tǒng)計(jì)的一般步驟這種不含任何未知參數(shù)的樣本的函數(shù)稱為統(tǒng)計(jì)量.它是完全由樣本決定的量.樣本是進(jìn)行統(tǒng)計(jì)推斷的依據(jù)。但在實(shí)際應(yīng)用時(shí)，一般不是直接使用樣本本身，而是對(duì)樣本進(jìn)行整理和加工,即針對(duì)具體問(wèn)題構(gòu)造適當(dāng)?shù)暮瘮?shù)—統(tǒng)計(jì)量,利用這些函數(shù)來(lái)進(jìn)行統(tǒng)計(jì)推斷,揭示總體的統(tǒng)計(jì)特性.一、統(tǒng)計(jì)量1.統(tǒng)計(jì)量的定義

定義6.2設(shè)X1，X2，…，Xn是來(lái)自總體X的樣本,x1，x2，…，xn為其樣本值,則稱不含任何總體分布中未知參數(shù)的連續(xù)函數(shù)為統(tǒng)計(jì)量,相應(yīng)實(shí)數(shù)稱為其觀察值。是不是實(shí)例12.幾個(gè)常用統(tǒng)計(jì)量(1)樣本平均值(2)(修正)樣本方差其觀察值(3)(修正)樣本標(biāo)準(zhǔn)差其觀察值其觀察值(4)樣本k階(原點(diǎn))矩其觀察值(5)樣本k階中心矩其觀察值證明定理6.1:設(shè)總體X的均值為μ,方差為σ2,(X1,X2,…,Xn)是X的一個(gè)樣本,則有證明辛欽定理再根據(jù)第五章辛欽定理知定理6.2:由第五章關(guān)于依概率收斂的序列的性質(zhì)知以上結(jié)論是下一章所要介紹的矩估計(jì)法的理論根據(jù).有關(guān)二維總體的統(tǒng)計(jì)量自己看。1.標(biāo)準(zhǔn)正態(tài)分布及其上側(cè)分位數(shù)若P(X>zα)=α,則稱zα為標(biāo)準(zhǔn)正態(tài)分布的上側(cè)α分位數(shù).zα

αXφ(x)其中定義設(shè)X~N(0,1),對(duì)任意0<α<1,二、常見抽樣分布完全由樣本確定的函數(shù)就是統(tǒng)計(jì)量。統(tǒng)計(jì)量是隨機(jī)變量，它的分布稱為抽樣分布。

下面,介紹來(lái)自正態(tài)總體的幾個(gè)重要統(tǒng)計(jì)量的分布.注:附表2-1附表2-2

定義

性質(zhì)

重要積分補(bǔ)充知識(shí):Γ-函數(shù)2.(卡方分布)的密度曲線Xf(x)n=1n=4n=10隨著n的增大,密度曲線逐漸趨于平緩,對(duì)稱.例1、設(shè)隨機(jī)變量X1，X２，X３，X４獨(dú)立且都服從N(0，1/2),則(X1+X2)2+(X3+X4)2服從_______分布;若要使aX12+b(X2+X3+X4)2~2(2),則

a=____,

b=____.

例2設(shè)是取自總體N(0,4)的簡(jiǎn)單隨機(jī)樣本

當(dāng)a=

,b=

時(shí),

解由題意得a=1/20b=1/100性質(zhì)1(此性質(zhì)可以推廣到多個(gè)隨機(jī)變量的情形.)性質(zhì)2證明Xf(x)附表附表3只詳列到n=45為止.附表附表例3例如利用上面公式,而查詳表可得費(fèi)舍爾()證明:t分布又稱學(xué)生氏(Student)分布.3.t分布的密度曲線:Xf(x)

特點(diǎn)

關(guān)于y軸對(duì)稱偶函數(shù);隨著自由度的逐漸增大,密度曲線逐漸接近于標(biāo)準(zhǔn)正態(tài)密度曲線.例4：設(shè)X1,X2,X3,X4是來(lái)自正態(tài)總體N(0,22)的簡(jiǎn)單隨機(jī)樣本，則服從______分布；Xf(x)α雙側(cè)α/2分位點(diǎn):顯然,由分布的對(duì)稱性知：附表3-1附表3-2例54.同理即：F分布的的密度函數(shù)的示意圖(n1,n2)=(10,40)(n1,n2)=(11,3)O例6：設(shè)X1,X2,X3,X4是來(lái)自正態(tài)總體N(0,22)的簡(jiǎn)單隨機(jī)樣本，則（Ｘ１２＋Ｘ２２）／（Ｘ３２＋Ｘ４２）服從______分布?！冀狻絫-分布,χ2-分布，F(xiàn)-分布。因?yàn)閄~t(n),所以由t-分布定義知：存在兩個(gè)相互獨(dú)立的隨機(jī)變量由Y,Z的相互獨(dú)立可得:Y2與Z也相互獨(dú)立。再由F-分布定義得:

使有

選擇題7P1506.已知，證明。由χ2-分布定義知：Xf(x)附表5-1附表5-2例7查附表6[P.301]:5.正態(tài)總體的樣本均值與樣本方差的分布對(duì)于單正態(tài)總體N(μ,σ2)的均值與方差有:

定理6.3設(shè)是來(lái)自正態(tài)總體N(μ,σ2)的樣本,則

①、

②、③、

④、獨(dú)立.注意:即2卡方分布定義證明且兩者獨(dú)立,由t分布的定義知解查表得則有由于定理6.4證明(1)由定理6.3(2)三、小結(jié)兩個(gè)最重要的統(tǒng)計(jì)量:樣本均值樣本方差三個(gè)來(lái)自正態(tài)分布的抽樣分布及其分位點(diǎn):〖解〗因?yàn)閄i~P(λ),所以E(Xi)=D(Xi)=λ(i=1,2,…,n),P1497.設(shè)X1，X2，X3，X4，X5為來(lái)自泊松分布P(λ)的一個(gè)樣本，為其樣本均值和（修正）樣本方差，求□例3-1〖解〗卡方分布及其數(shù)字特征。于是,由卡方分布數(shù)字特征知：由定理1知:

【練習(xí)】設(shè)在總體中抽取一容量為16的樣本，其中均為未知。（1）求概率；（2）求方差。(2)因?yàn)樗?□例4-續(xù)(1)P1505．設(shè)總體X~N（0，0.32）,n=10,求解：∵X/0.3~N（0，1），∴

【練習(xí)】在正態(tài)總體N(12,4)中隨機(jī)抽取容量為5的樣本X1，X2，X3，X4，X5，試求（1）樣本均值與總體均值之差的絕對(duì)值大于1的概率；（2）（3）〖解〗正態(tài)總體樣本均值的分布(1)因?yàn)樗杂谑?(2).(3).辛欽定理附表2-1標(biāo)準(zhǔn)正態(tài)分布表z01234567890.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.60.50000.53980.57930.61790.65540.69150.72570.75800.78810.81590.84130.86430.88490.90320.91920.93320.94520.50400.54380.58320.62170.65910.69500.72910.76110.79100.81860.84380.86650.88690.90490.92070.93450.94630.50800.54780.58710.62550.66280.69850.73240.76420.79390.82120.84610.86860.88880.90660.92220.93570.94740.51200.55170.59100.62930.66640.70190.73570.76730.79670.82380.84850.87080.89070.90820.92360.93700.94840.51600.55570.59480.63310.67000.70540.73890.77030.79950.82640.85080.87290.89250.90990.92510.93820.94950.51990.55960.59870.63680.67360.70880.74220.77340.80230.82890.85310.87490.89440.91150.92650.93940.95050.52390.56360.60260.64060.67720.71230.74540.77640.80510.83150.85540.87700.89620.91310.92780.94060.95150.52790.56750.60640.64430.68080.71570.74860.77940.80780.83400.85770.87900.89800.91470.92920.94180.95250.53190.57140.61030.64800.68440.71900.75170.78230.81060.83650.85990.88100.89970.91620.93060.94300.95350.53590.57530.61410.65170.68790.72240.75490.78520.81330.83890.86210.88300.90150.91770.93190.94410.95451.645附表2-2標(biāo)準(zhǔn)正態(tài)分布表z01234567891.61.71.81.92.02.12.22.32.42.52.62.72.82.93.00.94520.95540.96410.97130.97720.98210.98610.98930.99180.99380.99530.99650.99740.99810.99870.94630.95640.96480.97190.97780.98260.98640.98960.99200.99400.99550.99660.99750.99820.99900.94740.95730.96560.97260.97830.98300.98680.98980.99220.99410.99560.99670.99760.99820.99930.94840.95820.96640.97320.97880.98340.98710.99010.99250.99430.99570.99680.99770.99830.99950.94950.95910.96710.97380.97930.98380.98710.99040.99270.99450.99590.99690.99770.99840.99970.95050.95990.96780.97440.97980.98420.98780.99060.99290.99460.99600.99700.99780.99840.96980.95150.96080.96860.97500.98030.98460.98810.99090.99310.99480.99610.99710.99790.99850.99980.95250.96160.96930.97560.98080.98500.98840.99110.99320.99490.99620.99720.99790.99850.99990.95350.96250.97000.97620.98120.98540.98870.99130.99340.99510.99630.99730.99800.99860.99990.95450.96330.97060.97670.98170.98530.98900.99160.99360.99520.99640.99740.99810.99861.00001.96附表4-1=0.250.100.050.0250.010.005123456789101112131415161.3232.7734.1085.3856.6267.8419.03710.21911.38912.54913.70114.84515.98417.11718.24519.3692.7064.6056.2517.7799.23610.64512.01713.36214.68415.98717.27518.54919.81220.06422.30723.5423.8415.9917.8159.48811.07112.59214.06715.50716.91918.30719.67521.02622.36223.68524.99626.2965.0247.3789.34811.14312.83314.44916.01317.53519.02320.48321.92023.33724.73626.11927.48828.8456.6359.21011.34513.27715.08616.81218.47520.09021.66623.20924.72526.21727.68829.14130.57832.0007.87910.59712.83814.86016.75018.54820.27821.95523.58925.18826.75728.29929.89131.31932.80134.267分布表17.535=0.9950.990.9750.950.900.75123456789101112131415160.0100.0720.2070.4120.6760.9891.3441.7352.1562.6033.0743.5654.0754.6015.1420.0200.1150.2970.5540.8721.2391.6462.0882.5583.0533.5714.1074.6605.2295.8120.0010.0510.2160.4840.8311.2371.6902.1802.7003.2473.8164.4045.0095.6296.2626.9080.0040.1030.3520.7111.1451.6352.1672.7333.3253.9404.5755.2265.8926.5717.2617.9620.0160.2110.5841.0641.6102.2042.8333.4904.1684.8655.5786.3047.0427.7908.5479.3120.1020.5751.2131.9232.6753.4554.2555.0715.8996.7377.5848.4389.29910.16511.03711.9123.247附表4-2分布表=0.250.100.050.0250.010.0051718192021222324252627282930313220.48921.60522.71823.82824.93526.03927.14128.24129.33930.43531.52832.62033.71134.80035.88736.97324.76925.98927.20428.41229.61530.81332.00733.19634.38235.56336.74137.91639.08740.25641.42242.58527.58728.86930.14431.41032.67133.92435.17236.41537.65238.88540.11341.33742.55743.77344.98546.19430.19131.52632.85234.17035.47936.78138.07639.36440.64641.92343.19444.46145.71246.97948.23249.48033.40934.80536.19137.56638.93240.28941.63842.98044.31445.64246.96348.27849.58850.89252.19153.48635.71837.15638.58239.99741.40142.79644.18145.55946.92848.29049.64550.99352.33653.67255.00356.32834.382附表4-3分布表附表3-1=0.250.100.050.0250.010.005123456789101112131415161.00000.81650.76490.74070.72670.71760.71110.70640.70270.69980.69740.69550.69380.69240.69120.69013.07771.88561.63771.53321.47591.43981.41491.39681.38301.37221.36341.35621.35021.34501.34061.33686.31382.92002.35342.13182.01501.94321.89461.85951.83311.81251.79591.78231.77091.76131.75311.745912.70624.30273.18242.77642.57062.44692.36462.30602.26222.22812.20102.17882.16042.14482.13152.119931.82076.96464.54073.74693.36493.14272.99802.89652.82142.76382.71812.68102.65032.62452.60252.583563.65749.92485.84094.60414.03223.70743.49953.35543.24983.16933.10583.05453.01232.97682.94672.9208分布表1.8125附表3-2=0.250.100.050.0250.010.005123456789101112131415161.00000.81650.76490.74070.72670.71760.71110.70640.70270.69980.69740.69550.69380.69240.69120.69013.07771.88561.63771.53321.47591.43981.41491.39681.38301.37221.36341.35621.35021.34501.34061.33686.31382.92002.35342.13182.01501.94321.89461.85951.83311.81251.79591.78231.77091.76131.75311.745912.70624.30273.18242.77642.57062.44692.36462.30602.26222.22812.20102.17882.16042.14482.13152.119931.82076.96464.54073.74693.36493.14272.99802.89652.82142.76382.71812.68102.65032.62452.60252.583563.65749.92485.84094.60414.03223.70743.49953.35543.24983.16933.10583.05453.01232.97682.94672.92082.1315分布表附表5-1分布表

1234567891012152024304012012345678910111213141516171819647.838.5117.4412.2210.018.818.077.577.216.946.726.556.416.306.206.126.045.955.92799.539.0016.0410.658.437.266.546.065.715.465.265.104.974.864.774.694.624.564.51864.239.1715.449.987.766.605.895.425.084.834.634.474.354.244.154.084.013.953.90899.639.2515.109.607.396.235.525.504.724.474.284.124.003.893.803.733.663.613.56921.839.3014.889.367.155.995.294.824.484.244.043.893.773.663.583.503.443.383.33937.139.3314.739.206.985.825.124.654.234.073.883.733.603.503.413.343.283.223.17948.239.3614.629.076.855.704.994.534.203.953.763.613.483.383.293.223.163.103.05956.739.3714.548.986.765.604.904.434.103.853.663.513.393.293.203.123.063.012.96963.339.3914.478.906.685.524.824.364.033.783.593.443.313.213.123.052.982.932.88968.639.4014.428.846.625.464.764.303.963.723.533.373.253.153.062.992.922.872.82976.739.4114.348.756.525.374.674.203.873.623.433.283.153.052.962.892.822.772.72984.939.4314.258.666.435.274.574.103.773.523.333.183.052.952.862.792.722.672.62993.139.4514.178.566.335.174.474.003.673.423.233.072.952.842.762.682.622.562.51997.239.4614.128.516.285.124.423.593.613.373.173.022.892.792.702.632.562.502.45100139.4614.088.466.235.074.363.893.563.313.122.962.842.732.642.572.502.442.39100639.4714.048.416.185.014.313.843.513.263.062.912.782.672.592.512.442.382.33101439.4913.958.316.074.904.203.733.393.142.942.792.662.552.462.382.322.262.20101839.5013.908.266.024.854.143.673.333.082.882.722.602.492.402.322.252.192.134.53

1234567891015202430406012012345678910111213141516171819161.418.5110.137.716.615.995.595.323.124.964.844.754.674.604.544.494.454.414.38199.519.009.556.945.795.144.744.464.264.103.983.893.813.743.683.633.595.553.52215.719.169.286.595.414.764.354.073.813.713.593.493.413.343.293.243.203.163.13224.619.259.126.395.194.534.123.843.633.483.363.263.183.113.063.012.962.932.90230.219.309.016.265.054.393.973.693.48

3.333.203.113.032.962.902.852.812.772.74234.019.338.946.164.954.283.873.583.373.223.093.002.922.852.792.742.702.662.63236.819.358.896.094.884.213.793.503.293.143.012.912.832.762.712.662.612.582.54238.919.378.856.044.824.153.73