第一節點估計一、點估計問題的提法二、估計量的求法三、小結一、點估計問題的提法設總體X的分布函數形式已知,但它的一個或多個參數為未知,借助于總體X

的一個樣本來估計總體未知參數的值的問題稱為點估計問題.例1解由于用樣本均值依概率收斂于總體的均值，所以點估計問題的一般提法二、估計量的求法由于估計量是樣本的函數,是隨機變量,故對不同的樣本值,得到的參數值往往不同,如何求估計量是關鍵問題.常用構造估計量的方法:(兩種)矩估計法和最大似然估計法.1.矩估計法(X為連續型)(X為離散型)矩估計法的定義用樣本矩來估計總體矩,用樣本矩的連續函數來估計總體矩的連續函數,這種估計法稱為矩估計法.矩估計法的具體做法:矩估計量的觀察值稱為矩估計值.解根據矩估計法,例3解例4解方程組得到a,b的矩估計量分別為解例5解解方程組得到矩估計量分別為例6上例表明:

總體均值與方差的矩估計量的表達式不因不同的總體分布而異.一般地,2.最大似然估計法似然函數的定義最大似然估計法似然函數的定義求最大似然估計量的步驟:最大似然估計法是由費舍爾引進的.

最大似然估計法也適用于分布中含有多個未知參數的情況.此時只需令對數似然方程組對數似然方程解似然函數例7這一估計量與矩估計量是相同的.解例8這一估計量與矩估計量是相同的.解X的似然函數為例9它們與相應的矩估計量相同.解例10最大似然估計的性質U.證明

此性質可以推廣到總體分布中含有多個未知參數的情況.如例9中,三、小結兩種求點估計的方法:矩估計法最大似然估計法

在統計問題中往往先使用最大似然估計法,在最大似然估計法使用不方便時,再用矩估計法.第二節估計量的評選標準一、問題的提出二、無偏性三、有效性四、相合性五、小結一、問題的提出從前一節可以看到,對于同一個參數,用不同的估計方法求出的估計量可能不相同,如第一節的例4和例10.而且,很明顯,原則上任何統計量都可以作為未知參數的估計量.問題(1)對于同一個參數究竟采用哪一個估計量好?(2)評價估計量的標準是什么?下面介紹幾個常用標準.二、無偏性無偏估計的實際意義:無系統誤差.證例1特別的:不論總體X服從什么分布,只要它的數學期望存在,證例2(這種方法稱為無偏化).三、有效性由于方差是隨機變量取值與其數學期望的偏離程度,所以無偏估計以方差小者為好.四、相合性例如五、小結估計量的評選的三個標準無偏性有效性相合性相合性是對估計量的一個基本要求,不具備相合性的估計量是不予以考慮的.由最大似然估計法得到的估計量,在一定條件下也具有相合性.估計量的相合性只有當樣本容量相當大時,才能顯示出優越性,這在實際中往往難以做到,因此,在工程中往往使用無偏性和有效性這兩個標準.第四節區間估計一、區間估計的基本概念二、典型例題三、小結一、區間估計的基本概念1.置信區間的定義關于定義的說明若反復抽樣多次(各次得到的樣本容量相等,都是n)按伯努利大數定理,在這樣多的區間中,2.求置信區間的一般步驟(共3步)解例1二、典型例題這樣的置信區間常寫成其置信區間的長度為今抽9件測量其長度,得數據如下(單位:mm):142,138,150,165,156,148,132,135,160.解例2三、小結點估計不能反映估計的精度,故而本節引入了區間估計.求置信區間的一般步驟(分三步).第五節正態總體均值與方差的

區間估計一、單個總體的情況二、兩個總體的情況三、小結一、單個總體的情況由上節例2可知:1.包糖機某日開工包了12包糖,稱得質量(單位:克)分別為506,500,495,488,504,486,505,513,521,520,512,485.假設重量服從正態分布,解附表2-1例1附表2-2查表得推導過程如下:解有一大批糖果,現從中隨機地取16袋,稱得重量(克)如下:設袋裝糖果的重量服從正態分布,試求總體均值附表3-1例2就是說估計袋裝糖果重量的均值在500.4克與507.1克之間,這個估計的可信程度為95%.這個誤差的可信度為95%.解附表3-2例3(續例1)如果只假設糖包的重量服從正態分布解例4推導過程如下:根據第六章第二節定理二知2.進一步可得:注意:在密度函數不對稱時,習慣上仍取對稱的分位點來確定置信區間(如圖).

(續例2)求例2中總體標準差σ的置信度為0.95的置信區間.解代入公式得標準差的置信區間附表4-1附表4-2例5解例6

(續例1)二、兩個總體的情況討論兩個整體總體均值差和方差比的估計問題.推導過程如下:1.例7為比較?,??兩種型號步槍子彈的槍口速度,隨機地取?型子彈10發,得到槍口速度的平均值為隨機地取??型子彈20發,得槍口速度平均值為假設兩總體都可認為近似地服從正態分布,且由生產過程可認為它們的方差相等,求兩總體均值差信區間.解由題意,兩總體樣本獨立且方差相等(但未知),解由題意,兩總體樣本獨立且方差相等(但未知),例8為提高某一化學生產過程的得率,試圖采用一種新的催化劑,為慎重起見,在試驗工廠先進行體都可認為近似地服從正態分布,且方差相等,求兩總體均值差試驗.設采用原來的催化劑進行了次試驗,得到得率的平均值又采用新的催化劑進行了次試驗,得到得率的平均值假設兩總推導過程如下:2.根據F分布的定義,知解例9研究由機器A和機器B生產的鋼管內徑,隨機抽取機器A生產的管子18只,測得樣本方差為均未知,求方差比區間.設兩樣本相互獨抽取機器B生產的管子13只,測得樣本方差為立,且設由機器A和機器B生產的鋼管內徑分別服從正態分布信解例10甲、乙兩臺機床加工同一種零件,在機床甲加工的零件中抽取9個樣品,在機床乙加工的零件信區間.假定測量值都服從正態分布,方差分別為的置在置信度由所給數據算得0.98下,試求這兩臺機床加工精度之比中抽取6個樣品,并分別測得它們的長度(單位:mm),三、小結附表2-1標準正態分布表z01234567890.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.60.50000.53980.57930.61790.65540.69150.72570.75800.78810.81590.84130.86430.88490.90320.91920.93320.94520.50400.54380.58320.62170.65910.69500.72910.76110.79100.81860.84380.86650.88690.90490.92070.93450.94630.50800.54780.58710.62550.66280.69850.73240.76420.79390.82120.84610.86860.88880.90660.92220.93570.94740.51200.55170.59100.62930.66640.70190.73570.76730.79670.82380.84850.87080.89070.90820.92360.93700.94840.51600.55570.59480.63310.67000.70540.73890.77030.79950.82640.85080.87290.89250.90990.92510.93820.94950.51990.55960.59870.63680.67360.70880.74220.77340.80230.82890.85310.87490.89440.91150.92650.93940.95050.52390.56360.60260.64060.67720.71230.74540.77640.80510.83150.85540.87700.89620.91310.92780.94060.95150.52790.56750.60640.64430.68080.71570.74860.77940.80780.83400.85770.87900.89800.91470.92920.94180.95250.53190.57140.61030.64800.68440.71900.75170.78230.81060.83650.85990.88100.89970.91620.93060.94300.95350.53590.57530.61410.65170.68790.72240.75490.78520.81330.83890.86210.88300.90150.91770.93190.94410.95451.645z01234567891.61.71.81.92.02.12.22.32.42.52.62.72.82.93.00.94520.95540.96410.97130.97720.98210.98610.98930.99180.99380.99530.99650.99740.99810.99870.94630.95640.96480.97190.97780.98260.98640.98960.99200.99400.99550.99660.99750.99820.99900.94740.95730.96560.97260.97830.98300.98680.98980.99220.99410.99560.99670.99760.99820.99930.94840.95820.96640.97320.97880.98340.98710.99010.99250.99430.99570.99680.99770.99830.99950.94950.95910.96710.97380.97930.98380.98710.99040.99270.99450.99590.99690.99770.99840.99970.95050.95990.96780.97440.97980.98420.98780.99060.99290.99460.99600.99700.99780.99840.99980.95150.96080.96860.97500.98030.98460.98810.99090.99310.99480.99610.99710.99790.99850.99980.95250.96160.96930.97560.98080.98500.98840.99110.99320.99490.99620.99720.99790.99850.99990.95350.96250.97000.97620.98120.98540.98870.99130.99340.99510.99630.99730.99800.99860.99990.95450.96330.97060.97670.98170.98580.98900.99160.99360.99520.99640.99740.99810.99861.00001.96附表2-2標準正態分布表附表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2.1315=0.250.100.050.0250.010.005123456789101112131415161.00000.81650.76490.74070.72670.71760.71110.70640.70270.69980.69740.69550.69380.69240.69120.69013.07771.88561.63771.53321.47591.43981.41491.39681.38301.37221.36341.35621.35021.34501.34061.33686.31382.92002.35342.13182.01501.94321.89461.85951.83311.81251.79591.78231.77091.76131.75311.745912.70624.30273.18242.77642.57062.44692.36462.30602.26222.22812.20102.17882.16042.14482.13152.119931.82076.96464.54073.74693.36493.14272.99802.89652.82142.76382.71812.68102.65032.62452.60252.583563.65749.92485.84094.60414.03223.70743.49953.35543.24983.16933.10583.05453.01232.97682.94672.92082.2010附表3-2分布表附表4-2=0.9950.990.9750.950.900.75123456789101112131415160.0100.0720.2070.4120.6760.9891.3441.7352.1562.6033.0743.5654.0754.6015.1420.0200.1150.2970.5540.8721.2391.6462.0882.5583.0533.5714.1074.6605.2295.8120.0010.0510.2160.4840.8311.2371.6902.1802.7003.2473.8164.4045.0095.6296.2626.9080.0040.1030.3520.7111.1451.6352.1672.7333.3253.9404.5755.2265.8926.5717.2617.9620.0160.2110.5841.0641.6102.2042.8333.4904.1684.8655.5786.3047.0427.7908.5479.3120.1020.5751.2131.9232.6753.4554.2555.0715.8996.7377.5848.4389.29910.16511.03711.912分布表6.262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