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68 lines
1.9 KiB
Markdown
68 lines
1.9 KiB
Markdown
---
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title: Independence & Correlation
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tags:
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- math
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- statistics
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date: 2024-02-28
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---
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# Independent
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直白来说，独立（independence）指的是两个或多个变量之间的关系是否相互独立。如果两个变量是独立的，那么它们的取值不会相互影响。换句话说，一个变量的发生与其他变量的状态无关。例如，考虑一个骰子和一枚硬币，它们的投掷结果是独立的，因为你投掷硬币的结果不会影响骰子的结果。
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In mathematics，联合概率密度相当于分别的概率密度相乘
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$$
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p_{X,Y}(x,y) = p_X(x) * p_Y(y)
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$$
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$$
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\begin{equation}
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\begin{split} \rightarrow E[XY]
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& = \int\int xy p_{X,Y}(x,y)dxdy \\
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& =\int x p_X(x) \int y p_Y(y) \\
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& = E[X]E[Y]
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\end{split}
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\end{equation}
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$$
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# Correlation
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相关（correlation）指的是两个变量之间的关系是否存在关联。如果两个变量是相关的，那么它们的取值会彼此影响。当一个变量增加或减少时，另一个变量可能会相应地增加或减少。例如，考虑身高和体重之间的关系，一般来说，身高较高的人往往体重也较重。
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In mathematics, 一般使用correlation coefficient来判断二者的相关性，其表达式为：
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$$
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\rho(X, Y) = \frac{Cov[X,Y]}{\sqrt{Var[X] Var[Y]}}
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$$
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其中$Cov$ means Covariance，协方差:
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$$
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Cov[X, Y] = E[XY] - E[X]E[Y]
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$$
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所以不相关的表达为$\rho(X,Y) = 0$, 既$Cov[X,Y] = 0$, 也就是$E[XY] = E[X]E[Y]$
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# Conclusion
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* If $X$ and $Y$ are independent, they are also uncorrelated. **Independent -> Uncorrelated**, 既独立是强于不相关的束缚
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* 但是un-correlation无法推出independent
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## Math conclusion
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$$
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E[XY] = E[X]E[Y]
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$$
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-> un-correlation
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$$
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p_{X,Y}(x,y) = p_X(x) * p_Y(y)
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$$
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-> independent
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# Reference
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* [Uncorrelated-vs-independent.pdf](https://pinktalk.online/data_sci/basic/attachments/uncorrelated-vs-independent.pdf)
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