统计量与抽样分布概述

统计量与抽样分布概述总体选择个体样本观测样本样本观察值(数据)数据处理样本有关结论推断总体性质

统计量统计的一般步骤这种不含任何未知参数的样本的函数称为统计量.它是完全由样本决定的量.样本是进行统计推断的依据。但在实际应用时，一般不是直接使用样本本身，而是对样本进行整理和加工,即针对具体问题构造适当的函数—统计量,利用这些函数来进行统计推断,揭示总体的统计特性.一、统计量1.统计量的定义

定义6.2设X1，X2，…，Xn是来自总体X的样本,x1，x2，…，xn为其样本值,则称不含任何总体分布中未知参数的连续函数为统计量,相应实数称为其观察值。是不是实例12.几个常用统计量(1)样本平均值(2)(修正)样本方差其观察值(3)(修正)样本标准差其观察值其观察值(4)样本k阶(原点)矩其观察值(5)样本k阶中心矩其观察值证明定理6.1:设总体X的均值为μ,方差为σ2,(X1,X2,…,Xn)是X的一个样本,则有证明辛钦定理再根据第五章辛钦定理知定理6.2:由第五章关于依概率收敛的序列的性质知以上结论是下一章所要介绍的矩估计法的理论根据.有关二维总体的统计量自己看。1.标准正态分布及其上侧分位数若P(X>zα)=α,则称zα为标准正态分布的上侧α分位数.zα

αXφ(x)其中定义设X~N(0,1),对任意0<α<1,二、常见抽样分布完全由样本确定的函数就是统计量。统计量是随机变量，它的分布称为抽样分布。

下面,介绍来自正态总体的几个重要统计量的分布.注:附表2-1附表2-2

定义

性质

重要积分补充知识:Γ-函数2.(卡方分布)的密度曲线Xf(x)n=1n=4n=10随着n的增大,密度曲线逐渐趋于平缓,对称.例1、设随机变量X1，X２，X３，X４独立且都服从N(0，1/2),则(X1+X2)2+(X3+X4)2服从_______分布;若要使aX12+b(X2+X3+X4)2~2(2),则

a=____,

b=____.

例2设是取自总体N(0,4)的简单随机样本

当a=

,b=

时,

解由题意得a=1/20b=1/100性质1(此性质可以推广到多个随机变量的情形.)性质2证明Xf(x)附表附表3只详列到n=45为止.附表附表例3例如利用上面公式,而查详表可得费舍尔()证明:t分布又称学生氏(Student)分布.3.t分布的密度曲线:Xf(x)

特点

关于y轴对称偶函数;随着自由度的逐渐增大,密度曲线逐渐接近于标准正态密度曲线.例4：设X1,X2,X3,X4是来自正态总体N(0,22)的简单随机样本，则服从______分布；Xf(x)α双侧α/2分位点:显然,由分布的对称性知：附表3-1附表3-2例54.同理即：F分布的的密度函数的示意图(n1,n2)=(10,40)(n1,n2)=(11,3)O例6：设X1,X2,X3,X4是来自正态总体N(0,22)的简单随机样本，则（Ｘ１２＋Ｘ２２）／（Ｘ３２＋Ｘ４２）服从______分布。〖解〗t-分布,χ2-分布，F-分布。因为X~t(n),所以由t-分布定义知：存在两个相互独立的随机变量由Y,Z的相互独立可得:Y2与Z也相互独立。再由F-分布定义得:

使有

选择题7P1506.已知，证明。由χ2-分布定义知：Xf(x)附表5-1附表5-2例7查附表6[P.301]:5.正态总体的样本均值与样本方差的分布对于单正态总体N(μ,σ2)的均值与方差有:

定理6.3设是来自正态总体N(μ,σ2)的样本,则

①、

②、③、

④、独立.注意:即2卡方分布定义证明且两者独立,由t分布的定义知解查表得则有由于定理6.4证明(1)由定理6.3(2)三、小结两个最重要的统计量:样本均值样本方差三个来自正态分布的抽样分布及其分位点:〖解〗因为Xi~P(λ),所以E(Xi)=D(Xi)=λ(i=1,2,…,n),P1497.设X1，X2，X3，X4，X5为来自泊松分布P(λ)的一个样本，为其样本均值和（修正）样本方差，求□例3-1〖解〗卡方分布及其数字特征。于是,由卡方分布数字特征知：由定理1知:

【练习】设在总体中抽取一容量为16的样本，其中均为未知。（1）求概率；（2）求方差。(2)因为所以,□例4-续(1)P1505．设总体X~N（0，0.32）,n=10,求解：∵X/0.3~N（0，1），∴

【练习】在正态总体N(12,4)中随机抽取容量为5的样本X1，X2，X3，X4，X5，试求（1）样本均值与总体均值之差的绝对值大于1的概率；（2）（3）〖解〗正态总体样本均值的分布(1)因为所以于是,(2).(3).辛钦定理附表2-1标准正态分布表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标准正态分布表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