复旦概率论课件04数字特征和特征函数

整理文档很辛苦,赏杯茶钱您下走!

免费阅读已结束,点击下载阅读编辑剩下 ...

阅读已结束,您可以下载文档离线阅读编辑

资源描述

ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ11oŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇŒiA\£1⁄¢SflKI£2⁄'k(J£3⁄ŒiAl,¡N'…Œ#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇŒiA\£1⁄¢SflKI£2⁄'k(J£3⁄ŒiAl,¡N'…Œ#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇŒiA\£1⁄¢SflKI£2⁄'k(J£3⁄ŒiAl,¡N'…Œ#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇŒiA\£1⁄¢SflKI£2⁄'k(J£3⁄ŒiAl,¡N'…Œ#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇŒ˘ˇ‰´A.l.¯Cl¯CX'P(X=xn)=pn,n=1,2,...e?ŒP∞i=1xipiØ´æ§K¡T?ŒXŒ˘ˇ§PE(X)=∞Xi=1xipi#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇŒ˘ˇ‰´A.l.¯Cl¯CX'P(X=xn)=pn,n=1,2,...e?ŒP∞i=1xipiØ´æ§K¡T?ŒXŒ˘ˇ§PE(X)=∞Xi=1xipi#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~4.1.1X∼B(n,p),ƒE(X).):E(X)=nXk=0kCknpk(1−p)n−k=npnXk=1(n−1)!(k−1)!(n−k)!pk−1(1−p)(n−1)−(k−1)=npn−1Xk=0Ckn−1pk(1−p)(n−1)−k=npA~eY∼B(1,p),KE(Y)=p#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~4.1.1X∼B(n,p),ƒE(X).):E(X)=nXk=0kCknpk(1−p)n−k=npnXk=1(n−1)!(k−1)!(n−k)!pk−1(1−p)(n−1)−(k−1)=npn−1Xk=0Ckn−1pk(1−p)(n−1)−k=npA~eY∼B(1,p),KE(Y)=p#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~4.1.1X∼B(n,p),ƒE(X).):E(X)=nXk=0kCknpk(1−p)n−k=npnXk=1(n−1)!(k−1)!(n−k)!pk−1(1−p)(n−1)−(k−1)=npn−1Xk=0Ckn−1pk(1−p)(n−1)−k=npA~eY∼B(1,p),KE(Y)=p#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~4.1.1X∼B(n,p),ƒE(X).):E(X)=nXk=0kCknpk(1−p)n−k=npnXk=1(n−1)!(k−1)!(n−k)!pk−1(1−p)(n−1)−(k−1)=npn−1Xk=0Ckn−1pk(1−p)(n−1)−k=npA~eY∼B(1,p),KE(Y)=p#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~4.1.1X∼B(n,p),ƒE(X).):E(X)=nXk=0kCknpk(1−p)n−k=npnXk=1(n−1)!(k−1)!(n−k)!pk−1(1−p)(n−1)−(k−1)=npn−1Xk=0Ckn−1pk(1−p)(n−1)−k=npA~eY∼B(1,p),KE(Y)=p#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~4.1.1X∼B(n,p),ƒE(X).):E(X)=nXk=0kCknpk(1−p)n−k=npnXk=1(n−1)!(k−1)!(n−k)!pk−1(1−p)(n−1)−(k−1)=npn−1Xk=0Ckn−1pk(1−p)(n−1)−k=npA~eY∼B(1,p),KE(Y)=p#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇŒ˘ˇ‰´B.ºY5¯CºY¯CX…Œp(x),e¨'R∞−∞xp(x)dxØ´æ§K¡T¨'XŒ˘ˇ§PE(X)=Z∞−∞xp(x)dx#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇŒ˘ˇ‰´B.ºY5¯CºY¯CX…Œp(x),e¨'R∞−∞xp(x)dxØ´æ§K¡T¨'XŒ˘ˇ§PE(X)=Z∞−∞xp(x)dx#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~4.1.2X∼N(μ,σ2),ƒE(X).):E(X)=Z+∞−∞x1√2πσe−(x−μ)22σ2dxx−μσ=u=Z+∞−∞(uσ+μ)1√2πe−u22du=μ#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~4.1.2X∼N(μ,σ2),ƒE(X).):E(X)=Z+∞−∞x1√2πσe−(x−μ)22σ2dxx−μσ=u=Z+∞−∞(uσ+μ)1√2πe−u22du=μ#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~4.1.2X∼N(μ,σ2),ƒE(X).):E(X)=Z+∞−∞x1√2πσe−(x−μ)22σ2dxx−μσ=u=Z+∞−∞(uσ+μ)1√2πe−u22du=μ#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~4.1.2X∼N(μ,σ2),ƒE(X).):E(X)=Z+∞−∞x1√2πσe−(x−μ)22σ2dxx−μσ=u=Z+∞−∞(uσ+μ)1√2πe−u22du=μ#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~4.1.3•5¿:·⁄kr.v.kŒ˘ˇ~X(Cauchy)'…Œf(x)=1π(1+π2),−∞x+∞Z+∞−∞|x|f(x)dx=Z+∞−∞|x|π(1+π2)dxu§Œ˘ˇ3!#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~4.1.3•5¿:·⁄kr.v.kŒ˘ˇ~X(Cauchy)'…Œf(x)=1π(1+π2),−∞x+∞Z+∞−∞|x|f(x)dx=Z+∞−∞|x|π(1+π2)dxu§Œ˘ˇ3!#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇC./ξ'…ŒFξ(x)§KξŒ˘ˇ‰´XeE(ξ)=ZR1xdFξ(x).øp¨'RR1xdFξ(x)…Œf(x)=x’u'…ŒFξ(x)Riemanm−Stieltjes¨'§‰´XeZ∞−∞xdFξ(x),limΔ→0Xixi(Fξ(xi+1)−Fξ(xi))#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇC./ξ'…ŒFξ(x)§KξŒ˘ˇ‰´XeE(ξ)=ZR1xdFξ(x).øp¨'RR1xdFξ(x)…Œf(x)=x’u'…ŒFξ(x)Riemanm−Stieltjes¨'§‰´XeZ∞−∞xdFξ(x),limΔ→0Xixi(Fξ(xi+1)−Fξ(xi))#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇC./ξ'…ŒFξ(x)§KξŒ˘ˇ‰´XeE(ξ)=ZR1xdFξ(x).øp¨'RR1xdFξ(x)…Œf(x)=x’u'…ŒFξ(x)Riemanm−Stieltjes¨'§‰´XeZ∞−∞xdFξ(x),limΔ→0Xixi(Fξ(xi+1)−Fξ(xi))#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇC./ξ'…ŒFξ(x)§KξŒ˘ˇ‰´XeE(ξ)=ZR1xdFξ(x).øp¨'RR1xdFξ(x)…Œf(x)=x’u'…ŒFξ(x)Riemanm−Stieltjes¨'§‰´XeZ∞−∞xdFξ(x),limΔ→0Xixi(Fξ(xi+1)−Fξ(xi))#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇC./ξ'…ŒFξ(x)§KξŒ˘ˇ‰´XeE(ξ)=ZR1xdFξ(x).øp¨'RR1xdFξ(x)…Œf(x)=x’u'…ŒFξ(x)Riemanm−Stieltjes¨'§‰´XeZ∞−∞xdFξ(x),limΔ→0Xixi(Fξ(xi+1)−Fξ(xi))#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇC./ξ'…ŒFξ(x)§KξŒ˘ˇ‰´XeE(ξ)=ZR1xdFξ(x).øp¨'RR1xdFξ(x)…Œf(x)=x’u'…ŒFξ(x)Riemanm−Stieltjes¨'§‰´XeZ∞−∞xdFξ(x),limΔ→0Xixi(Fξ(xi+1)−Fξ(xi))#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇC./ξ'…ŒFξ(x)§KξŒ˘ˇ‰´XeE(ξ)=ZR1xdFξ(x).øp¨'RR1xdFξ(x)…Œf(x)=x’u'…ŒFξ(x)Riemanm−Stieltjes¨'§‰´XeZ∞−∞xdFξ(x),limΔ→0Xixi(Fξ(xi+1)−Fξ(xi))#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~'Œ˘ˇA'ξ∼G(p)§KE(ξ)=∞Xi=1iqi−1p=p×ddx(1−x)−1 x=1−p=1p#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~'Œ˘ˇA'ξ∼G(p)§KE(ξ)=∞Xi=1iqi−1p=p×ddx(1−x)−1 x=1−p=1p#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~'Œ˘ˇA'ξ∼G(p)§KE(ξ)=∞Xi=1iqi−1p=p×ddx(1−x)−1 x=1−p=1p#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~'Œ˘ˇ˚Ct'ξ∼P(λ)§KE(ξ)=∞Xi=0i×λii!e−λ=λ!'ξ∼U(a,b)§KE(ξ)=Zbaxb−adx=a+b2#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~'Œ˘ˇ˚Ct'ξ∼P(λ)§KE(ξ)=∞Xi=0i×λii!e−λ=λ!'ξ∼U(a,b)§KE(ξ)=Zbaxb−adx=a+b2#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~'Œ˘ˇΓ-'ξ∼Γ(λ,r)§KξŒ˘ˇE(ξ)=Z∞0x·λrΓ(r)xr−1e−λxdx=rλZ∞0λr+1Γ(r+1)x(r+1)−1e−λxdx=rλ.#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~'Œ˘ˇΓ-'ξ∼Γ(λ,r)§KξŒ˘ˇE(ξ)=Z∞0x·λrΓ(r)xr−1e−λxdx=rλZ∞0λr+1Γ(r+1)x(r+1)−1e−λxdx=rλ.#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ~'Œ˘ˇΓ-'ξ∼Γ(λ,r)§KξŒ˘ˇE(ξ)=Z∞0x·λrΓ(r)xr−1e−λxdx=rλZ∞0λr+1Γ(r+1)x(r+1)−1e−λxdx=rλ.#)V˙˜:ŒiAA…ŒŒ˘ˇ¯CIO^Œ˘ˇ¯C…Œ

1 / 400
下载文档,编辑使用

©2015-2020 m.777doc.com 三七文档.

备案号:鲁ICP备2024069028号-1 客服联系 QQ:2149211541

×
保存成功