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概率论与数理统计中的独立(独立独立同分布不相关)

發(fā)布時間：2024/3/26 编程问答 38 豆豆

生活随笔收集整理的這篇文章主要介紹了概率论与数理统计中的独立(独立独立同分布不相关) 小編覺得挺不錯的,現(xiàn)在分享給大家,幫大家做個參考.

$由均值方差的性質(zhì),Z=x?μσ2，則E(z)=0,var(z)=1由均值方差的性質(zhì),Z=\frac{x- μ}{\sqrt{σ^2}}，則E(z)=0,var(z)=1$

關(guān)鍵詞：獨立獨立同分布不相關(guān)

獨立

$P(A*B)=P(A)*P(B)\\ 二維：p_{ij}=p_{i·}*p_{·j}$
$(獨立與相容：互不相融AB=Φ)\tiny ( 獨立與相容：互不相融 AB=Φ )$

不相關(guān)

$不相關(guān) 性質(zhì) ： E （ X, Y ） = E (X) E (Y)$

$相關(guān)與斜方差Cov：ρxy=Cov(x,y)var(x)var(y)∑(xt?xˉ)(yt?yˉ)∑(xt?xˉ)2∑(yt?yˉ)2=(xt?xˉ→)?(yt?yˉ→)∣xt?xˉ→∣?∣yt?yˉ→∣去均值內(nèi)積相關(guān)與斜方差Cov：\\ \rho_{xy}=\frac{Cov(x,y)}{\sqrt{var(x)var(y)}} \\ \frac{\sum(x_t-\bar x)(y_t-\bar y)}{\sum(x_t-\bar x)^2\sum(y_t-\bar y)^2}= \frac{(\overrightarrow{x_t-\bar x} ) \cdot (\overrightarrow{y_t-\bar y})}{|\overrightarrow{x_t-\bar x}|\cdot |\overrightarrow{y_t-\bar y}|} \color{red}去均值內(nèi)積$

性質(zhì)(類比向量的點乘)：

$（1）Cov(X，Y)=Cov(Y，X)；（2）Cov(aX，bY)=abCov(X，Y)，（a，b是常數(shù)）；（3）Cov(X1+X2，Y)=Cov(X1，Y)+Cov(X2，Y)。所以有：Cov(X+Y,X?Y)=Cov(X,X)?Cov(Y,Y)\tiny（1）Cov(X，Y)=Cov(Y，X)；\\ （2）Cov(aX，bY)=abCov(X，Y)，（a，b是常數(shù)）；\\ （3）Cov(X1+X2，Y)=Cov(X1，Y)+Cov(X2，Y)。\\ 所以有： Cov(X+Y,X-Y)=Cov(X,X)-Cov(Y,Y)$

計算（類比方差）：

$Cov(X,Y)=E(XY)?E(X)E(Y)D(X+Y)=D(X)+D(Y)+2Cov(X，Y)D(X?Y)=D(X)+D(Y)?2Cov(X，Y)\tiny Cov(X,Y)=E(XY)-E(X)E(Y) \\ D(X+Y)=D(X)+D(Y)+2Cov(X，Y) \\ D(X-Y)=D(X)+D(Y)-2Cov(X，Y) \\$

正交

$正交性質(zhì) ： E （ X, Y ） = E (X) E (Y) = 0$
$X, Y 不相關(guān) + 有一個均值為 0 = 正交$

$例：X,Y服從單位圓內(nèi)的均勻分布，不相關(guān)，不獨立\color{red} 例：X,Y服從單位圓內(nèi)的均勻分布，不相關(guān)，不獨立$

獨立同分布

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.edge-pattern-dashed{stroke-dasharray:3}#mermaid-svg-PQsxSrL6dgDRy1O2 .edge-pattern-dotted{stroke-dasharray:2}#mermaid-svg-PQsxSrL6dgDRy1O2 .marker{fill:#333}#mermaid-svg-PQsxSrL6dgDRy1O2 .marker.cross{stroke:#333}:root { --mermaid-font-family: "trebuchet ms", verdana, arial;}#mermaid-svg-PQsxSrL6dgDRy1O2 {color: rgba(0, 0, 0, 0.75);font: ;}數(shù)理統(tǒng)計中的-獨立- *大數(shù)定律 *中心極限定理 *抽樣分布 *均要求獨立同分布 *

大數(shù)定律-百度百科
中心極限定理-百度百科-獨立同分布的中心極限定理

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