假设 X X X 和 Y Y Y 是两个随机变量,我们需要求 X Y XY XY 的方差。
方差公式:
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\text{Var}(X) = \mathbb{E}[(X - E(X))^2] = \mathbb{E}[(X^2 - 2XE(X) + E(X))^2] = \mathbb{E}[X^2] - (\mathbb{E}[X])^2
Var(X)=E[(X−E(X))2]=E[(X2−2XE(X)+E(X))2]=E[X2]−(E[X])2
根据方差的定义,方差是平方值的期望减去期望值的平方,即
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\text{Var}(XY) = \mathbb{E}[(XY)^2] - (\mathbb{E}[XY])^2
Var(XY)=E[(XY)2]−(E[XY])2
求 E [ X Y ] \mathbb{E}[XY] E[XY]:
根据协方差定义:
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\text{Cov}(X, Y) = \mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])]
Cov(X,Y)=E[(X−E[X])(Y−E[Y])]
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= \mathbb{E}[XY - X\mathbb{E}[Y] - \mathbb{E}[X]Y + \mathbb{E}[X]\mathbb{E}[Y]]
=E[XY−XE[Y]−E[X]Y+E[X]E[Y]]
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= \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y] - \mathbb{E}[X]\mathbb{E}[Y] + \mathbb{E}[X]\mathbb{E}[Y]
=E[XY]−E[X]E[Y]−E[X]E[Y]+E[X]E[Y]
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= \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y]
=E[XY]−E[X]E[Y]
有: E [ X Y ] = E [ X ] E [ Y ] + Cov ( X , Y ) \mathbb{E}[XY] = \mathbb{E}[X] \mathbb{E}[Y] + \text{Cov}(X,Y) E[XY]=E[X]E[Y]+Cov(X,Y)
求 E [ ( X Y ) 2 ] \mathbb{E}[(XY)^2] E[(XY)2]:
E [ ( X Y ) 2 ] = E [ X 2 Y 2 ] \mathbb{E}[(XY)^2] = \mathbb{E}[X^2 Y^2] E[(XY)2]=E[X2Y2]
同样我们可以利用协方差公式得到:
E [ X 2 Y 2 ] = E [ X 2 ] E [ Y 2 ] + Cov ( X 2 , Y 2 ) \mathbb{E}[X^2 Y^2] = \mathbb{E}[X^2] \mathbb{E}[Y^2] + \text{Cov}(X^2, Y^2) E[X2Y2]=E[X2]E[Y2]+Cov(X2,Y2)
将其代入:
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\mathbb{E}[(XY)^2] = \mathbb{E}[X^2 Y^2]
E[(XY)2]=E[X2Y2]
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\mathbb{E}[(XY)^2] = \mathbb{E}[X^2] \mathbb{E}[Y^2] + \text{Cov}(X^2, Y^2)
E[(XY)2]=E[X2]E[Y2]+Cov(X2,Y2)
求 Var ( X Y ) \text{Var}(XY) Var(XY) :
Var ( X Y ) = E [ X 2 ] E [ Y 2 ] + Cov ( X 2 , Y 2 ) − ( E [ X ] E [ Y ] + Cov ( X , Y ) ) 2 \text{Var}(XY) = \mathbb{E}[X^2] \mathbb{E}[Y^2] + \text{Cov}(X^2, Y^2) - (\mathbb{E}[X] \mathbb{E}[Y] + \text{Cov}(X,Y))^2 Var(XY)=E[X2]E[Y2]+Cov(X2,Y2)−(E[X]E[Y]+Cov(X,Y))2
如果两个变量独立,进一步简化
假设 X X X 和 Y Y Y 是两个独立的随机变量,可以推导出 X 2 X^2 X2和 Y 2 Y^2 Y2也是独立的。
推导 X 2 X^2 X2和 Y 2 Y^2 Y2独立:
如果 X X X 和 Y Y Y 是独立的,则 X X X 和 Y Y Y 的联合概率密度函数可以分解为各自的概率密度函数的乘积:
f X , Y ( x , y ) = f X ( x ) f Y ( y ) f_{X,Y}(x, y) = f_X(x) f_Y(y) fX,Y(x,y)=fX(x)fY(y)
在这种情况下,我们有:
E [ X Y ] = E [ X ] E [ Y ] \mathbb{E}[XY] = \mathbb{E}[X] \mathbb{E}[Y] E[XY]=E[X]E[Y]
但这并不直接意味着 E [ X 2 Y 2 ] = E [ X 2 ] E [ Y 2 ] \mathbb{E}[X^2 Y^2] = \mathbb{E}[X^2] \mathbb{E}[Y^2] E[X2Y2]=E[X2]E[Y2]。