線代與概率論第五章

第五章

數(shù)理統(tǒng)計的基本概念

與抽樣分布第5.1節(jié)基本概念一、總體與個體二、隨機樣本三、統(tǒng)計量四、小結(jié)一、總體與個體一個統(tǒng)計問題總有它明確的研究對象.研究對象的全體稱為總體(母體)，總體中每個成員稱為個體.研究某批燈泡的質(zhì)量…考察國產(chǎn)轎車的質(zhì)量總體總體然而在統(tǒng)計研究中，人們往往關(guān)心每個個體的一項(或幾項)數(shù)量指標(biāo)和該數(shù)量指標(biāo)在總體中的分布情況.這時，每個個體具有的數(shù)量指標(biāo)的全體就是總體.該批燈泡壽命的全體就是總體燈泡的壽命國產(chǎn)轎車每公里的耗油量所有國產(chǎn)轎車每公里耗油量的全體就是總體由于每個個體的出現(xiàn)帶有隨機性，即相應(yīng)的數(shù)量指標(biāo)值的出現(xiàn)帶有隨機性。從而可把此種數(shù)量指標(biāo)看作隨機變量，我們用一個隨機變量或其分布來描述總體。為此常用隨機變量的符號或分布的符號來表示總體。

通常，我們用隨機變量X,Y,Z,…,等表示總體。當(dāng)我們說到總體，就是指一個具有確定概率分布的隨機變量。如:研究某批燈泡的壽命時，我們關(guān)心的數(shù)量指標(biāo)就是壽命，那么，此總體就可以用隨機變量X表示，或用其分布函數(shù)F(x)表示.總體某批燈泡的壽命壽命X可用一概率分布來刻劃F(x)某工廠10月份生產(chǎn)的燈泡壽命所組成的總體中,個體的總數(shù)就是10月份生產(chǎn)的燈泡數(shù),這是一個有限總體;而該工廠生產(chǎn)的所有燈泡壽命所組成的總體可近似地看成一個無限總體,它包括以往生產(chǎn)和今后生產(chǎn)的燈泡壽命.

有限總體和無限總體實例當(dāng)有限總體包含的個體的總數(shù)很大時,可近似地將它看成是無限總體.二、隨機樣本1.樣本的定義為推斷總體的分布及各種特征,按一定的規(guī)則從總體中抽取若干個體進行觀察試驗,以獲得有關(guān)總體的信息.這一抽取過程稱為“抽樣”.所抽取的部分個體稱為樣本.通常記為樣本中所包含的個體數(shù)目n稱為樣本容量.容量為n的樣本可以看作n維隨機變量.但是,一旦取定一組樣本,得到的是n個具體的數(shù),稱此為樣本的一次觀察值,簡稱樣本值.2.簡單隨機樣本抽取樣本的目的是為了利用樣本對總體進行統(tǒng)計推斷,這就要求樣本能很好的反映總體的特性且便于處理.為此,需對抽樣提出一些要求,通常有兩條:滿足上述兩條性質(zhì)的樣本稱為簡單隨機樣本.獲得簡單隨機樣本的抽樣方法稱為簡單隨機抽樣.為了使大家對總體和樣本有一個明確的概念,我們給出如下定義:定義5.1一個隨機變量X或其相應(yīng)的分布函數(shù)F(x)稱為一個總體.1.代表性：X1,X2,…,Xn中每一個與所考察的總體X有相同的分布.2.獨立性：X1,X2,…,Xn是相互獨立的隨機變量.定義5.2樣本所有可能取值的全體稱為樣本空間，記為。定理5.13.樣本的分布解例1解例2三、統(tǒng)計量1.

統(tǒng)計量的定義5.3由樣本推斷總體特征,需要對樣本進行“加工”,“提煉”.這就需要構(gòu)造一些樣本的函數(shù),它把樣本中所含的信息集中起來.是不是例12.

幾個常用統(tǒng)計量(樣本矩)的定義(1)樣本平均值(2)樣本方差其觀察值它反映了總體均值的信息它反映了總體方差的信息其觀察值(3)樣本標(biāo)準(zhǔn)差其觀察值(4)修正樣本方差(5)

樣本k階(原點)矩其觀察值(6)樣本k階中心矩其觀察值其觀察值樣本矩具有下列性質(zhì):性質(zhì)5.1

證明證明再根據(jù)第四章辛欽定理知性質(zhì)5.2由第四章關(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法的理論根據(jù).

四、小結(jié)個體總體有限總體無限總體基本概念:說明1

一個總體對應(yīng)一個隨機變量X,我們將不區(qū)分總體和相應(yīng)的隨機變量,統(tǒng)稱為總體X.說明2

在實際中遇到的總體往往是有限總體,它對應(yīng)一個離散型隨機變量;當(dāng)總體中包含的個體的個數(shù)很大時,在理論上可認(rèn)為它是一個無限總體.隨機樣本總體,樣本,樣本值的關(guān)系總體(理論分布)樣本樣本值?數(shù)理統(tǒng)計是從手中已有的資料--樣本值,去推斷總體的情況--總體的分布F(x)的性質(zhì).樣本是聯(lián)系二者的橋梁.總體分布決定了樣本取值的概率規(guī)律,也就是樣本取到樣本值的規(guī)律,因而可以由樣本值去推斷總體.兩個最重要的統(tǒng)計量:樣本均值樣本方差第5.2節(jié)常用統(tǒng)計分布一、常見分布二、概率分布的分位數(shù)三、小結(jié)一、性質(zhì)1(此性質(zhì)可以推廣到多個隨機變量的情形)性質(zhì)2證明同理與相互獨立補充:概率分布的分位數(shù)附表2-1根據(jù)正態(tài)分布的對稱性知附表2-2附表4-1附表4只詳列到n=45為止.附表4-2附表4-3例如利用上公式,而查詳表可得費歇(R.A.Fisher)證明:t分布又稱學(xué)生氏(Student)分布.二、當(dāng)n充分大時,其圖形類似于標(biāo)準(zhǔn)正態(tài)變量概率密度的圖形.由分布的對稱性知附表3-1附表3-2三、其中分別表示F分布的分位數(shù)附表5-1附表5-2證明三、小結(jié)1.三大統(tǒng)計分布的定義,性質(zhì).2.概率分布的分位數(shù)辛欽定理附表2-1標(biāo)準(zhǔn)正態(tài)分布表z0.000.010.020.030.040.050.060.070.080.090.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.60.50000.53980.57930.61790.65540.69150.72570.75800.78810.81590.84130.86430.88490.90320.91920.93320.94520.50400.54380.58320.62170.65910.69500.72910.76110.79100.81860.84380.86650.88690.90490.92070.93450.94630.50800.54780.58710.62550.66280.69850.73240.76420.79390.82120.84610.86860.88880.90660.92220.93570.94740.51200.55170.59100.62930.66640.70190.73570.76730.79670.82380.84850.87080.89070.90820.92360.93700.94840.51600.55570.59480.63310.67000.70540.73890.77030.79950.82640.85080.87290.89250.90990.92510.93820.94950.51990.55960.59870.63680.67360.70880.74220.77340.80230.82890.85310.87490.89440.91150.92650.93940.95050.52390.56360.60260.64060.67720.71230.74540.77640.80510.83150.85540.87700.89620.91310.92780.94060.95150.52790.56750.60640.64430.68080.71570.74860.77940.80780.83400.85770.87900.89800.91470.92920.94180.95250.53190.57140.61030.64800.68440.71900.75170.78230.81060.83650.85990.88100.89970.91620.93060.94300.95350.53590.57530.61410.65170.68790.72240.75490.78520.81330.83890.86210.88300.90150.91770.93190.94410.95451.645附表2-2標(biāo)準(zhǔn)正態(tài)分布表z0.000.010.020.030.040.050.060.070.080.091.61.71.81.92.02.12.22.32.42.52.62.72.82.93.00.94520.95540.96410.97130.97720.98210.98610.98930.99180.99380.99530.99650.99740.99810.99870.94630.95640.96480.97190.97780.98260.98640.98960.99200.99400.99550.99660.99750.99820.99900.94740.95730.96560.97260.97830.98300.98680.98980.99220.99410.99560.99670.99760.99820.99930.94840.95820.96640.97320.97880.98340.98710.99010.99250.99430.99570.99680.99770.99830.99950.94950.95910.96710.97380.97930.98380.98710.99040.99270.99450.99590.99690.99770.99840.99970.95050.95990.96780.97440.97980.98420.98780.99060.99290.99460.99600.99700.99780.99840.96980.95150.96080.96860.97500.98030.98460.98810.99090.99310.99480.99610.99710.99790.99850.99980.95250.96160.96930.97560.98080.98500.98840.99110.99320.99490.99620.99720.99790.99850.99990.95350.96250.97000.97620.98120.98540.98870.99130.99340.99510.99630.99730.99800.99860.99990.95450.96330.97060.97670.98170.98530.98900.99160.99360.99520.99640.99740.99810.99861.00001.96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

1234567891012152024304012012345678910111213141516171819647.838.5117.4412.2210.018.818.077.577.216.946.726.556.416.306.206.126.045.955.92799.539.0016.0410.658.437.266.546.065.715.465.265.104.974.864.774.694.624.564.51864.239.1715.449.987.766.605.895.425.084.834.634.474.354.244.154.084.013.953.90899.639.2515.109.607.396.235.525.504.724.474.284.124.003.893.803.733.663.613.56921.839.3014.889.367.155.995.294.824.484.244.043.893.773.663.583.503.443.383.33937.139.3314.739.206.985.825.124.654.234.073.883.733.603.503.413.343.283.223.17948.239.3614.629.076.855.704.994.534.203.953.763.613.483.383.293.223.163.103.05956.739.3714.548.986.765.604.904.434.103.853.663.513.393.293.203.123.063.012.96963.339.3914.478.906.685.524.824.364.033.783.593.443.313.213.123.052.982.932.88968.639.4014.428.846.625.464.764.303.963.723.533.373.253.153.062.992.922.872.82976.739.4114.348.756.525.374.674.203.873.623.433.283.153.052.962.892.822.772.72984.939.4314.258.666.435.274.574.103.773.523.333.183.052.952.862.792.722.672.62993.139.4514.178.566.335.174.474.003.673.423.233.072.952.842.762.682.622.562.51997.239.4614.128.516.285.124.423.593.613.373.173.022.892.792.702.632.562.502.45100139.4614.088.466.235.074.363.893.563.313.122.962.842.732.642.572.502.442.39100639.4714.048.416.185.014.313.843.513.263.062.912.782.672.592.512.442.382.33101439.4913.958.316.074.904.203.733.393.142.942.792.662.552.462.382.322.262.20101839.5013.908.266.024.854.143.673.333.082.882.722.602.492.402.322.252.192.134.90

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

3.333.203.113.032.962.902.852.812.772.74234.019.338.946.164.954.283.873.583.373.223.093.002.922.852.792.742.702.662.63236.819.358.896.094.884.213.793.503.293.143.012.912.832.762.712.662.612.582.54238.919.378.856.044.824.153.733.443.233.072.952.852.772.702.642.592.552.512.4824.0519.388.816.004.774.103.683.393.183.022.902.802.712.652.592.542.492.462.42241.919.408.795.964.744.063.643.353.142.982.852.752.672.602.542.492.452.412.38245.919.438.705.864.623.942.513.223.012.852.722.622.532.462.402.352.312.272.23248.019.458.665.804.563.87