確率変数の和（差）の分散の一般公式の導出

（I）分散の定義式 $V\left(X\right)=E\left[{\{X-E\left(X\right)\}}^2\right]$ より
（i）和の分散 $$\begin{array}{r}V(X+Y)=E\left[{\{(X+Y)-E(X+Y)\}}^2\right]\end{array}$$ 期待値の性質 $E(X+Y)=E\left(X\right)+E\left(Y\right)$ より、 $$\begin{array}{rl}V(X+Y)& =E\left[{\{X+Y-E\left(X\right)-E\left(Y\right)\}}^2\right]\\ & =E\left[{[\{X-E\left(X\right)\}+\{Y-E\left(Y\right)\}]}^2\right]\\ & =E[{\{X-E\left(X\right)\}}^2+2\{X-E\left(X\right)\}\{Y-E\left(Y\right)\}+{\{Y-E\left(Y\right)\}}^2]\end{array}$$ 期待値の性質 $E(X+Y)=E\left(X\right)+E\left(Y\right)$ より、 $$\begin{array}{r}V(X+Y)=E\left[{\{X-E\left(X\right)\}}^2\right]+2E\left[\{X-E\left(X\right)\}\{Y-E\left(Y\right)\}\right]+E\left[{\{Y-E\left(Y\right)\}}^2\right]\end{array}$$ 分散と共分散の定義式より、 $$\begin{array}{r}V(X+Y)=V\left(X\right)+V\left(Y\right)+2\mathrm{Cov}(X,Y)\end{array}$$

（ii）差の分散 $$\begin{array}{r}V(X-Y)=E\left[{\{(X-Y)-E(X-Y)\}}^2\right]\end{array}$$ 期待値の性質 $E(X-Y)=E\left(X\right)-E\left(Y\right)$ より、 $$\begin{array}{rl}V(X-Y)& =E\left[{\{X-Y-E\left(X\right)+E\left(Y\right)\}}^2\right]\\ & =E\left[{[\{X-E\left(X\right)\}-\{Y-E\left(Y\right)\}]}^2\right]\\ & =E[{\{X-E\left(X\right)\}}^2-2\{X-E\left(X\right)\}\{Y-E\left(Y\right)\}+{\{Y-E\left(Y\right)\}}^2]\end{array}$$ 期待値の性質 $E(X+Y)=E\left(X\right)+E\left(Y\right)$ より、 $$\begin{array}{r}V(X+Y)=E\left[{\{X-E\left(X\right)\}}^2\right]-2E\left[\{X-E\left(X\right)\}\{Y-E\left(Y\right)\}\right]+E\left[{\{Y-E\left(Y\right)\}}^2\right]\end{array}$$ 分散と共分散の定義式より、 $$\begin{array}{r}V(X-Y)=V\left(X\right)+V\left(Y\right)-2\mathrm{Cov}(X,Y)\end{array}$$ $◼$

（II）分散の定義式 $V\left(X\right)=E\left[{\{X-E\left(X\right)\}}^2\right]$ より $$\begin{array}{rl}V\left(\sum _{i=1}^n{X}_i\right)& =E\left[{\left(\sum _{i=1}^n\{X_i-E\left(X_i\right)\}\right)}^2\right]\\ & =E\left[\sum _{j=1}^n\sum _{i=1}^n\{X_i-E\left(X_i\right)\}\{X_j-E\left(X_j\right)\}\right]\\ & =\sum _{j=1}^n\sum _{i=1}^{n}E\left[\{X_i-E\left(X_i\right)\}\{X_j-E\left(X_j\right)\}\right]\\ & =\sum _{j=1}^n\sum _{i=1}^n\mathrm{Cov}(X_i,X_j)\end{array}$$ ここで、$\mathrm{Cov}(X_i,X_i)=V\left(X_i\right),\mathrm{Cov}(X_i,X_j)=\mathrm{Cov}(X_j,X_i)$ なので、 $$\begin{array}{rl}V\left(\sum _{i=1}^n{X}_i\right)& =\sum _{i=1}^{n}V\left(X_i\right)+\sum _{j\ne i}\sum _{i=1}^n\mathrm{Cov}(X_i,X_j)\\ & =\sum _{i=1}^{n}V\left(X_i\right)+2\sum \sum _{i<j}\mathrm{Cov}(X_i,X_j)\end{array}$$ $◼$