평균과 분산 성질

\(Var(X) = E[(X - \mu)^{2}] \)

\(= \underset{x}{\Sigma} (x-\mu)^{2} p(x) \)

\(= \underset{x}{\Sigma} (x^{2}-2\mu x +\mu^{2}) p(x) \)

\(= \underset{x}{\Sigma} x^{2}p(x)-2\mu \underset{x}{\Sigma} xp(x) +\mu^{2} \underset{x}{\Sigma} p(x) \)

\(= E[X^{2}] - 2\mu^{2} + \mu^{2}\)

\(= E[X^{2}] - \mu^{2}\)

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\(E[X+Y] = E[X] + E[Y]\)

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\(E[aX + b] = aE[X] + b = a\mu + b \)

 

\(Var(aX + b) = E[(aX + b - a\mu - b)^{2}] = a^{2}E[(X - \mu)^{2}] = a^{2}Var(X)\)

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Theorem: If \(X\) & \(Y\) are independent, then \(E[XY] = E[X]E[Y]\)

Proof:

Let \(x_{i}, y_{i}, i = 1, 2, \ldots\) be the possible values of \(X, Y\).

 

\(E[XY] = \underset{i}{\Sigma}\underset{j}{\Sigma} x_{i}y_{j}P(X = x_{i} \wedge Y = y_{j}) \) \( = \underset{i}{\Sigma}\underset{j}{\Sigma} x_{i}y_{j}P(X = x_{i}) P(Y = y_{j}) \) \( = \underset{i}{\Sigma}x_{i}P(X=x_{i})\underset{j}{\Sigma} y_{j}P(Y = y_{j}) \) \( = E[X]E[Y] \)

 

Note: Not true in general;

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In general: \(Var[X+Y] \ne Var[X] + Var[Y]\)

Theorem: If \(X\) & \(Y\) are independent, then \(Var[X+Y] = Var[x]+Var[y]\)

Proof:

\( \hat{X}= X - E[X], \hat{Y}= Y - E[Y] \) 
\( E[\hat{X}] = 0, E[\hat{Y}] = 0 \)
\( Var[\hat{X}] = Var[X], Var[\hat{Y}] = Var[Y] \)

 

\( Var[X + Y] = Var[\hat{X} + \hat{Y}]\) 
\( = E[(\hat{X} + \hat{Y})^{2}]- (E[\hat{X} + \hat{Y}])^{2}\)
\( = E[\hat{X}^{2} + 2\hat{X})\hat{Y} + \hat{Y}^{2}]- 0\)
\( = E[\hat{X}^{2}] + 2E[\hat{X})\hat{Y}] + E[\hat{Y}^{2}]\)
\( = Var[\hat{X}] + 0 + Var[\hat{Y}]\)
\( = Var[X] + Var[Y]\)

 

\( Std[X + Y] = \sqrt{Var[X] + Var[Y]}\)