共分散の定義と基本性質の証明

恒等式 $$\begin{array}{r}{[\{X-E\left(X\right)\}-\{Y-E\left(Y\right)\}]}^2\ge 0\end{array}$$ を変形すると、 $$\begin{array}{c}{\{X-E\left(X\right)\}}^2-2\{X-E\left(X\right)\}\{Y-E\left(Y\right)\}+{\{Y-E\left(Y\right)\}}^2\ge 0\\ \{X-E\left(X\right)\}\{Y-E\left(Y\right)\}\le \frac{1}{2}[{\{X-E\left(X\right)\}}^2+{\{Y-E\left(Y\right)\}}^2]\end{array}$$ 両辺の期待値を取ると、 $$\begin{array}{r}E\left[\{X-E\left(X\right)\}\{Y-E\left(Y\right)\}\right]\le \frac{1}{2}E[{\{X-E\left(X\right)\}}^2+{\{Y-E\left(Y\right)\}}^2]\end{array}$$ 分散と共分散の定義式より、 $$\begin{array}{r}\mathrm{Cov}(X,Y)\le \frac{1}{2}E[V\left(X\right)+V\left(Y\right)]<\mathrm{\infty }\end{array}$$ $◼$

共分散の定義式 $\mathrm{Cov}(X,Y)=E\left[\{X-E\left(X\right)\}\{Y-E\left(Y\right)\}\right]$ の右辺の中身を展開すると、 $$\begin{array}{r}\mathrm{Cov}(X,Y)=E\{XY-XE\left(Y\right)-YE\left(X\right)+E\left(X\right)E\left(Y\right)\}\end{array}$$ $E\left(X\right)$ と $E\left(Y\right)$ は定数なので、期待値の性質 $E\left(a\right)=a,E(aX+b)=aE\left(X\right)+b$ より、 $$\begin{array}{rl}\mathrm{Cov}(X,Y)& =E\left(XY\right)-E\left(X\right)E\left(Y\right)-E\left(X\right)E\left(Y\right)+E\left(X\right)E\left(Y\right)\\ & =E\left(XY\right)-E\left(X\right)E\left(Y\right)\end{array}$$ $◼$

共分散の公式 $\mathrm{Cov}(X,Y)=E\left(XY\right)-E\left(X\right)E\left(Y\right)$ より、 $$\begin{array}{r}\mathrm{Cov}(aX+b,cY+d)=E\left\{(aX+b)(cY+d)\right\}-E(aX+b)E(cY+d)\end{array}$$ 期待値の性質 $E\left(a\right)=a,E(aX+b)=aE\left(X\right)+b$ より、 $$\begin{array}{rl}\mathrm{Cov}(aX+b,cY+d)& =E(acXY+adX+bcY+bd)-\{aE\left(X\right)+b\}\{cE\left(Y\right)+d\}\\ & =acE\left(XY\right)+adE\left(X\right)+bcE\left(Y\right)+bd-\{acE\left(X\right)E\left(Y\right)+adE\left(X\right)+bcE\left(Y\right)+bd\}\\ & =ac\{E\left(XY\right)-E\left(X\right)E\left(Y\right)\}\end{array}$$ 共分散の公式から、$\mathrm{Cov}(X,Y)=E\left(XY\right)-E\left(X\right)E\left(Y\right)$ なので、 $$\begin{array}{r}\mathrm{Cov}(aX+b,cY+d)=ac\cdot \mathrm{Cov}(X,Y)\end{array}$$ $◼$

共分散の定義式に従って $X+Y$ と $Z$ の共分散を求めると、 $$\begin{array}{rl}\mathrm{Cov}(X+Y,Z)& =E\left[\{(X+Y)-E(X+Y)\}\{Z-E\left(Z\right)\}\right]\\ & =E[Z(X+Y)-E\left(Z\right)(X+Y)-E(X+Y)Z+E(X+Y)E\left(Z\right)]\\ & =E[XZ+YZ-E\left(Z\right)X-E\left(Z\right)Y-\{E\left(X\right)+E\left(Y\right)\}Z+\{E\left(X\right)+E\left(Y\right)\}E\left(Z\right)]\\ & =E\left(XZ\right)+E\left(YZ\right)-E\left(X\right)E\left(Z\right)-E\left(Y\right)E\left(Z\right)-E\left(X\right)E\left(Z\right)-E\left(Y\right)E\left(Z\right)+E\left(X\right)E\left(Z\right)+E\left(Y\right)E\left(Z\right)\\ & =E\left(XZ\right)-E\left(X\right)E\left(Z\right)+E\left(YZ\right)-E\left(Y\right)E\left(Z\right)\end{array}$$ 共分散の公式 $\mathrm{Cov}(X,Y)=E\left(XY\right)-E\left(X\right)E\left(Y\right)$ より、 $$\begin{array}{r}\mathrm{Cov}(X+Y,Z)=\mathrm{Cov}(X,Z)+\mathrm{Cov}(Y,Z)\end{array}$$ $◼$