數(shù)理統(tǒng)計(jì)

1、11統(tǒng)計(jì)量與抽樣分布統(tǒng)計(jì)量與抽樣分布1.1基本概念：統(tǒng)計(jì)量、樣本矩、經(jīng)驗(yàn)分布函數(shù)基本概念：統(tǒng)計(jì)量、樣本矩、經(jīng)驗(yàn)分布函數(shù)總體X的樣本X1，X2，…，Xn，則T(X1，X2，…，Xn)即為統(tǒng)計(jì)量樣本均值X??樣本方差212)(1????niinXXnS修正樣本方差212)(11?????niinXXnS樣本k階原點(diǎn)矩...)21(11????kXnAnikik樣本k階中心矩...)21()(11?????kXXnBnikik經(jīng)驗(yàn)分布函數(shù)其中

2、Vn(x)表示隨機(jī)事件出現(xiàn)的)()()(???????xnxvxFnnxX?次數(shù)顯然則有))((~)(xFnBxVn)()]([xFxFEn?)](1)[(1)]([xFxFnxFDn??補(bǔ)充：補(bǔ)充：?DXnnESn12??DXESn?222)(EXDXEX???22211nniiSXXn?????二項(xiàng)分布B(np):)...10()1(nkppCkXPknkkn?????EX=npDX=np(1p)?泊松分布:)(?P...)10(!

3、????kekkXPk????EX??DX?均勻分布U(ab):)(1)(bxaabxf????2baEX??2)(121abDX???指數(shù)分布:()(0)()1(0)xxfxexFxex????????????1?EX21??DX?正態(tài)分布:)(2??N2)(exp21)(22???????xxf??EX2??DX3?的概率密度XYZ?dxxxzxfzfz??????)()(?的概率密度)(xgy?)]([))(()(11ygygfy

4、fxy????函數(shù)：????????01)(dxexx??)()1(???????1)1()!1()(?????nn?B函數(shù)：?????1011)1()(dxxxB????)()()()(???????????B1.4次序統(tǒng)計(jì)量及其分布：次序統(tǒng)計(jì)量、樣本中位數(shù)次序統(tǒng)計(jì)量及其分布：次序統(tǒng)計(jì)量、樣本中位數(shù)、樣本極差、樣本極差RAXX(k)的分布密度：)...21()()](1[)]([)!()!1(!)(1)(nkxfxFxFknknxfk

5、nkxk???????X(1)的分布密度：1)](1)[()()1(???nxxFxnfxfX(n)的分布密度：1)]()[()()(??nxxFxnfxfn2參數(shù)估計(jì)參數(shù)估計(jì)2.1點(diǎn)估計(jì)與優(yōu)良性：概念、無(wú)偏估計(jì)、均方誤差準(zhǔn)則、相合估計(jì)點(diǎn)估計(jì)與優(yōu)良性：概念、無(wú)偏估計(jì)、均方誤差準(zhǔn)則、相合估計(jì)(一致估計(jì)一致估計(jì))、漸近正、漸近正態(tài)估計(jì)態(tài)估計(jì)的均方誤差：??????22()()()MSEEDE????????????若是無(wú)偏估計(jì)，則????(

6、)MSED????對(duì)于的任意一個(gè)無(wú)偏估計(jì)量，有，則是的最小方差無(wú)偏估計(jì)，記?????DD??????MVUE相合估計(jì)(一致估計(jì)):limnnE??????lim0nnD????2.2點(diǎn)估計(jì)量的求法：矩估計(jì)法、最大似然估計(jì)法點(diǎn)估計(jì)量的求法：矩估計(jì)法、最大似然估計(jì)法矩估計(jì)法：1求出總體的k階原點(diǎn)矩:12(...)kkkmaEXxdFx??????????2解方程組(k=12...m)，得即為所求11nkkiiaXn?????12(...)k