第二節統計量與抽樣分布一、基本概念二、常見分布三、小結第二節統計量與抽樣分布一、基本概念二、常見分布三、小結1總體選擇個體樣本觀測樣本樣本觀察值(數據)數據處理樣本有關結論推斷總體性質

統計量統計的一般步驟這種不含任何未知參數的樣本的函數稱為統計量.它是完全由樣本決定的量.樣本是進行統計推斷的依據。但在實際應用時，一般不是直接使用樣本本身，而是對樣本進行整理和加工,即針對具體問題構造適當的函數—統計量,利用這些函數來進行統計推斷,揭示總體的統計特性.總體選擇個體樣本觀測樣本樣本觀察值(數據)數據處理樣本有關結2一、統計量1.

統計量的定義

定義6.2設X1，X2，…，Xn是來自總體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的連續函數為統計量,相應實數稱為其觀察值。一、統計量1.統計量的定義定義6.2設X1，X2，…，3是不是實例1是不是實例142.

幾個常用統計量(1)樣本平均值(2)(修正)樣本方差其觀察值2.幾個常用統計量(1)樣本平均值(2)(修正)樣本方差其5(3)(修正)樣本標準差其觀察值其觀察值(3)(修正)樣本標準差其觀察值其觀察值6(4)

樣本k階(原點)矩其觀察值(5)樣本k階中心矩其觀察值(4)樣本k階(原點)矩其觀察值(5)樣本k階中心7證明

定理6.1:設總體X的均值為μ,方差為σ2,(X1,X2,…,Xn)是X的一個樣本,則有證明定理6.1:設總體X的均值為μ,方差為σ2,(X1,X8概率論_抽樣分布課件9證明辛欽定理再根據第五章辛欽定理知定理6.2:證明辛欽定理再根據第五章辛欽定理知定理6.2:10由第五章關于依概率收斂的序列的性質知以上結論是下一章所要介紹的矩估計法的理論根據.有關二維總體的統計量自己看。由第五章關于依概率收斂的序列的性質知以上結論111.標準正態分布及其上側分位數若P(X>zα)=α,則稱zα為標準正態分布的上側α分位數.zα

αXφ(x)其中定義設X~N(0,1),對任意0<α<1,二、常見抽樣分布完全由樣本確定的函數就是統計量。統計量是隨機變量，它的分布稱為抽樣分布。

下面,介紹來自正態總體的幾個重要統計量的分布.1.標準正態分布及其上側分位數若P(X>zα)=α,zαα12注:注:13附表2-1附表2-2附表2-1附表2-214

定義

性質

重要積分補充知識:Γ-函數定義性質重要積分補充知識:Γ-函數152.(卡方分布)2.(卡方分布)16

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,對稱.的密度曲線Xf(x)n=1n=4n=17例1、設隨機變量X1，X２，X３，X４獨立且都服從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1、設隨機變量X1，X２，X３，X４獨立且都服從N(0，118

例2設是取自總體N(0,4)的簡單隨機樣本

當a=

,b=

時,

解由題意得a=1/20b=1/100例2設是取自總體N19性質1(此性質可以推廣到多個隨機變量的情形.)性質1(此性質可以推廣到多個隨機變量的情形.)20性質2證明性質2證明21Xf(x)Xf(x)22附表附表3只詳列到n=45為止.附表附表例3附表附表3只詳列到n=45為止.附表附表例323例如利用上面公式,而查詳表可得費舍爾(R.A.Fisher)證明:例如利用上面公式,而查詳表可得費舍爾(R.A.Fisher)24t分布又稱學生氏(Student)分布.3.t分布又稱學生氏(Student)分布.3.25t分布的密度曲線:Xf(x)

特點

關于y軸對稱偶函數;隨著自由度的逐漸增大,密度曲線逐漸接近于標準正態密度曲線.t分布的密度曲線:Xf(x)特點關于y軸對稱偶函數;26概率論_抽樣分布課件27例4：設X1,X2,X3,X4是來自正態總體N(0,22)的簡單隨機樣本，則服從______分布；例4：設X1,X2,X3,X4是來自正態總體N(0,22)的28Xf(x)αXf(x)α29雙側α/2分位點:顯然,由分布的對稱性知：雙側α/2分位點:顯然,由分布的對稱性知：30附表3-1附表3-2例5附表3-1附表3-2例5314.同理即：4.同理即：32概率論_抽樣分布課件33F分布的的密度函數的示意圖(n1,n2)=(10,40)(n1,n2)=(11,3)OF分布的的密度函數的示意圖(n1,n2)=(10,40)(n34例6：設X1,X2,X3,X4是來自正態總體N(0,22)的簡單隨機樣本，則（Ｘ１２＋Ｘ２２）／（Ｘ３２＋Ｘ４２）服從______分布。例6：設X1,X2,X3,X4是來自正態總體N(0,22)的35〖解〗t-分布,χ2-分布，F-分布。

因為X~t(n),所以由t-分布定義知：存在兩個相互獨立的隨機變量

由Y,Z的相互獨立可得:Y2與Z也相互獨立。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：〖解〗t-分布,χ2-分布，F-分布。因36Xf(x)Xf(x)37附表5-1附表5-2例7附表5-1附表5-2例738概率論_抽樣分布課件39查附表6[P.301]:查附表6[P.301]:405.正態總體的樣本均值與樣本方差的分布對于單正態總體N(μ,σ2)的均值與方差有:

定理6.3設是來自正態總體N(μ,σ2)的樣本,則

①、

②、

③、

④、獨立.注意:即2卡方分布定義5.正態總體的樣本均值與樣本方差的分布對于單正態總41證明且兩者獨立,由t

分布的定義知證明且兩者獨立,由t分布的定義知42解查表得則有由于解查表得則有由于43定理6.4定理6.444概率論_抽樣分布課件45證明(1)由定理6.3證明(1)由定理6.346(2)(2)47概率論_抽樣分布課件48三、小結兩個最重要的統計量:樣本均值樣本方差三個來自正態分布的抽樣分布及其分位點:三、小結兩個最重要的統計量:樣本均值樣本方差三個來自正態分布49〖解〗因為Xi~P(λ),所以E(Xi)=D(Xi)=λ(i=1,2,…,n),P1497.設X1，X2，X3，X4，X5為來自泊松分布P(λ)的一個樣本，

為其樣本均值和（修正）樣本方差，求〖解〗因為Xi~P(λ),所以E(Xi)=D(Xi)=λ(i50□例3-1□例3-151〖解〗卡方分布及其數字特征

。于是,由卡方分布數字特征知：由定理1知:

【練習】

設在總體

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

。

〖解〗卡方分布及其數字特征。于是,由卡方分布數字特征知：由52(2)因為所以,□例4-續(1)(2)因為所以,□例4-續(1)53P1505．設總體X~N（0，0.32）,n=10,求解：∵X/0.3~N（0，1），∴P1505．設總體X~N（0，0.32）,n=1054

【練習】在正態總體N(12,4)中隨機抽取容量為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值大于1的概率；（2）

（3）

〖解〗正態總體樣本均值的分布

(1)因為

所以

于是,【練習】在正態總體N(12,4)中隨機抽取容量為5的樣本55(2).(3).(2).(3).56辛欽定理辛欽定理57附表2-1標準正態分布表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2-1標準正態分布表z01234567890.00.5058附表2-2標準正態分布表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2-2標準正態分布表z01234567891.60.9459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附表4-1=0.250.100.050.0250.010.060=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=0.9950.990.9750.950.900.7510.61=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=0.250.100.050.0250.010.00517262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3-1=0.250.100.050.0250.010.63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3-2=0.250.100.050.0250.010.64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附表5-1分布表1234567891065

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

3.333.203.113.032.962.902.852.812.772.74234.019.338.946.164.954.283.873.583.373.223.093.002.922.852.792.742.702.662.63236.819.358.896.094.884.213.793.503.293.143.012.912.832.762.712.662.612.582.54238.919.378.856.044.824.153.733.443.233.072.952.852.772.702.642.592.552.512.4824.0519.388.816.004.774.103.683.393.183.022.902.802.712.652.592.542.492.462.42241.919.408.795.964.744.063.643.353.142.982.852.752.672.602.542.492.452.412.38245.919.438.705.864.623.942.513.223.012.852.722.622.532.462.402.352.312.272.23248.019.458.665.804.563.873.443.152.942.772.652.542.462.392.332.282.232.192.16249.119.458.645.774.533.843.413.122.902.742.612.512.422.352.292.242.192.152.11250.119.468.625.754.503.813.383.082.86

2.702.575.472.382.312.252.192.152.112.07151.119.478.595.724.463.773.343.042.832.662.532.432.342.272.202.152.102.062.03252.219.488.575.694.433.743.303.012.792.622.492.382.302.22

2.162.112.062.021.98253.319.498.555.664.403.703.272.972.752.582.452.342.252.182.112.062.011.971.93254.319.508.535.634.363.673.232.932.712.542.402.302.212.132.072.011.961.921.882.31附表5-2分布表123456789101520243066說明

(修正)樣本方差還可表示為【推導】說明(修正)樣本方差還可表示為【推導】67第二節統計量與抽樣分布一、基本概念二、常見分布三、小結第二節統計量與抽樣分布一、基本概念二、常見分布三、小結68總體選擇個體樣本觀測樣本樣本觀察值(數據)數據處理樣本有關結論推斷總體性質

統計量統計的一般步驟這種不含任何未知參數的樣本的函數稱為統計量.它是完全由樣本決定的量.樣本是進行統計推斷的依據。但在實際應用時，一般不是直接使用樣本本身，而是對樣本進行整理和加工,即針對具體問題構造適當的函數—統計量,利用這些函數來進行統計推斷,揭示總體的統計特性.總體選擇個體樣本觀測樣本樣本觀察值(數據)數據處理樣本有關結69一、統計量1.

統計量的定義

定義6.2設X1，X2，…，Xn是來自總體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的連續函數為統計量,相應實數稱為其觀察值。一、統計量1.統計量的定義定義6.2設X1，X2，…，70是不是實例1是不是實例1712.

幾個常用統計量(1)樣本平均值(2)(修正)樣本方差其觀察值2.幾個常用統計量(1)樣本平均值(2)(修正)樣本方差其72(3)(修正)樣本標準差其觀察值其觀察值(3)(修正)樣本標準差其觀察值其觀察值73(4)

樣本k階(原點)矩其觀察值(5)樣本k階中心矩其觀察值(4)樣本k階(原點)矩其觀察值(5)樣本k階中心74證明

定理6.1:設總體X的均值為μ,方差為σ2,(X1,X2,…,Xn)是X的一個樣本,則有證明定理6.1:設總體X的均值為μ,方差為σ2,(X1,X75概率論_抽樣分布課件76證明辛欽定理再根據第五章辛欽定理知定理6.2:證明辛欽定理再根據第五章辛欽定理知定理6.2:77由第五章關于依概率收斂的序列的性質知以上結論是下一章所要介紹的矩估計法的理論根據.有關二維總體的統計量自己看。由第五章關于依概率收斂的序列的性質知以上結論781.標準正態分布及其上側分位數若P(X>zα)=α,則稱zα為標準正態分布的上側α分位數.zα

αXφ(x)其中定義設X~N(0,1),對任意0<α<1,二、常見抽樣分布完全由樣本確定的函數就是統計量。統計量是隨機變量，它的分布稱為抽樣分布。

下面,介紹來自正態總體的幾個重要統計量的分布.1.標準正態分布及其上側分位數若P(X>zα)=α,zαα79注:注:80附表2-1附表2-2附表2-1附表2-281

定義

性質

重要積分補充知識:Γ-函數定義性質重要積分補充知識:Γ-函數822.(卡方分布)2.(卡方分布)83

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,對稱.的密度曲線Xf(x)n=1n=4n=84例1、設隨機變量X1，X２，X３，X４獨立且都服從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1、設隨機變量X1，X２，X３，X４獨立且都服從N(0，185

例2設是取自總體N(0,4)的簡單隨機樣本

當a=

,b=

時,

解由題意得a=1/20b=1/100例2設是取自總體N86性質1(此性質可以推廣到多個隨機變量的情形.)性質1(此性質可以推廣到多個隨機變量的情形.)87性質2證明性質2證明88Xf(x)Xf(x)89附表附表3只詳列到n=45為止.附表附表例3附表附表3只詳列到n=45為止.附表附表例390例如利用上面公式,而查詳表可得費舍爾(R.A.Fisher)證明:例如利用上面公式,而查詳表可得費舍爾(R.A.Fisher)91t分布又稱學生氏(Student)分布.3.t分布又稱學生氏(Student)分布.3.92t分布的密度曲線:Xf(x)

特點

關于y軸對稱偶函數;隨著自由度的逐漸增大,密度曲線逐漸接近于標準正態密度曲線.t分布的密度曲線:Xf(x)特點關于y軸對稱偶函數;93概率論_抽樣分布課件94例4：設X1,X2,X3,X4是來自正態總體N(0,22)的簡單隨機樣本，則服從______分布；例4：設X1,X2,X3,X4是來自正態總體N(0,22)的95Xf(x)αXf(x)α96雙側α/2分位點:顯然,由分布的對稱性知：雙側α/2分位點:顯然,由分布的對稱性知：97附表3-1附表3-2例5附表3-1附表3-2例5984.同理即：4.同理即：99概率論_抽樣分布課件100F分布的的密度函數的示意圖(n1,n2)=(10,40)(n1,n2)=(11,3)OF分布的的密度函數的示意圖(n1,n2)=(10,40)(n101例6：設X1,X2,X3,X4是來自正態總體N(0,22)的簡單隨機樣本，則（Ｘ１２＋Ｘ２２）／（Ｘ３２＋Ｘ４２）服從______分布。例6：設X1,X2,X3,X4是來自正態總體N(0,22)的102〖解〗t-分布,χ2-分布，F-分布。

因為X~t(n),所以由t-分布定義知：存在兩個相互獨立的隨機變量

由Y,Z的相互獨立可得:Y2與Z也相互獨立。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：〖解〗t-分布,χ2-分布，F-分布。因103Xf(x)Xf(x)104附表5-1附表5-2例7附表5-1附表5-2例7105概率論_抽樣分布課件106查附表6[P.301]:查附表6[P.301]:1075.正態總體的樣本均值與樣本方差的分布對于單正態總體N(μ,σ2)的均值與方差有:

定理6.3設是來自正態總體N(μ,σ2)的樣本,則

①、

②、

③、

④、獨立.注意:即2卡方分布定義5.正態總體的樣本均值與樣本方差的分布對于單正態總108證明且兩者獨立,由t

分布的定義知證明且兩者獨立,由t分布的定義知109解查表得則有由于解查表得則有由于110定理6.4定理6.4111概率論_抽樣分布課件112證明(1)由定理6.3證明(1)由定理6.3113(2)(2)114概率論_抽樣分布課件115三、小結兩個最重要的統計量:樣本均值樣本方差三個來自正態分布的抽樣分布及其分位點:三、小結兩個最重要的統計量:樣本均值樣本方差三個來自正態分布116〖解〗因為Xi~P(λ),所以E(Xi)=D(Xi)=λ(i=1,2,…,n),P1497.設X1，X2，X3，X4，X5為來自泊松分布P(λ)的一個樣本，

為其樣本均值和（修正）樣本方差，求〖解〗因為Xi~P(λ),所以E(Xi)=D(Xi)=λ(i117□例3-1□例3-1118〖解〗卡方分布及其數字特征

。于是,由卡方分布數字特征知：由定理1知:

【練習】

設在總體

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

。

〖解〗卡方分布及其數字特征。于是,由卡方分布數字特征知：由119(2)因為所以,□例4-續(1)(2)因為所以,□例4-續(1)120P1505．設總體X~N（0，0.32）,n=10,求解：∵X/0.3~N（0，1），∴P1505．設總體X~N（0，0.32）,n=10121

【練習】在正態總體N(12,4)中隨機抽取容量為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值大于1的概率；（2）

（3）

〖解〗正態總體樣本均值的分布

(1)因為

所以

于是,【練習】在正態總體N(12,4)中隨機抽取容量為5的樣本122(2).(3).(2).(3).123辛欽定理辛欽定理124附表2-1標準正態分布表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2-1標準正態分布表z01234567890.00.50125附表2-2標準正態分布表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2-2標準正態分布表z01234567891.60.94126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附表4-1=0.250.100.050.0250.010.0127=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=0.9950.990.9750.950.900.7510.128=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=0.250.100.050.0250.010.005172129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3-1=0.250.100.050.0250.010.130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.