$3$\displaystyle{V\left({ aX+bY }\right)}$ $3$\displaystyle{={a}^{2}V\left({X}\right)+2abC\left({ X,Y }\right)+{b}^{2}V\left({Y}\right)}$

2つの確率変数 $3$\displaystyle{X}$ ， $3$\displaystyle{Y}$ について

$3$\displaystyle{V\left({ aX+bY }\right)}$ $3$\displaystyle{={a}^{2}V\left({X}\right)+2abC\left({ X,Y }\right)+{b}^{2}V\left({Y}\right)}$ ･･････(1)

が成り立つ．

■証明

$3$\displaystyle{E\left({X}\right)=\hspace{1}{{\mu}}_{x}}$ ･･････(2)

$3$\displaystyle{E\left({Y}\right)=\hspace{1}{{\mu}}_{y}}$ ･･････(3)

とする．

分散の定義より

$3$\displaystyle{V\left({ aX+bY }\right)}$ $3$\displaystyle{=E\left({ { \left{{ \left({ aX+bY }\right)- \left({ a\hspace{1}{{\mu}}_{x}+b\hspace{1}{{\mu}}_{y} }\right) }\right} }^{2} }\right)}$

$3$\displaystyle{=E\left({ { \left({ aX+bY }\right) }^{2} }\right)- {\left{{ E\left({ a\hspace{1}{{\mu}}_{x}+b\hspace{1}{{\mu}}_{y} }\right) }\right}}^{2}}$

∵ $3$\displaystyle{V\left({X}\right)=E\left({ {X}^{2} }\right)- {\left{{ E\left({X}\right) }\right}}^{2}}$ (分散を参照)

$3$\displaystyle{=E\left({ {a}^{2}{X}^{2}+2abXY+{b}^{2}{Y}^{2} }\right)- {\left({ a\hspace{1}{{\mu}}_{x}+b\hspace{1}{{\mu}}_{y} }\right)}^{2}}$

$3$\displaystyle{=E\left({ {a}^{2}{X}^{2} }\right)+E\left({ 2abXY }\right)+E\left({ {b}^{2}{Y}^{2} }\right)}$ $3$\displaystyle{- {a}^{2}\hspace{1}{{\mu}}_{x}{ }^{2}}$ $3$\displaystyle{- 2ab\hspace{1}{{\mu}}_{x}\hspace{1}{{\mu}}_{y}}$ $3$\displaystyle{- {b}^{2}\hspace{1}{{\mu}}_{y}{ }^{2}}$

$3$\displaystyle{={a}^{2}E\left({ {X}^{2} }\right)- {a}^{2}\hspace{1}{{\mu}}_{x}{ }^{2}+2abE\left({ XY }\right)}$ $3$\displaystyle{- 2ab\hspace{1}{{\mu}}_{x}\hspace{1}{{\mu}}_{y}}$ $3$\displaystyle{+{b}^{2}E\left({ {Y}^{2} }\right)}$ $3$\displaystyle{- {b}^{2}\hspace{1}{{\mu}}_{y}{ }^{2}}$

$3$\displaystyle{={a}^{2}\left{{ E\left({ {X}^{2} }\right)- \hspace{1}{{\mu}}_{x}{ }^{2} }\right}}$ $3$\displaystyle{+2ab\left{{ E\left({ XY }\right)- \hspace{1}{{\mu}}_{x}\hspace{1}{{\mu}}_{y} }\right}}$ $3$\displaystyle{ +{b}^{2}\left{{ E\left({ {Y}^{2} }\right)- \hspace{1}{{\mu}}_{y}{ }^{2} }\right} }$

$3$\displaystyle{={a}^{2}\left[{ E\left({ {X}^{2} }\right)- { \left{{ E\left({X}\right) }\right} }^{2} }\right]}$ $3$\displaystyle{+2ab\left{{ E\left({ XY }\right)- E\left({X}\right)E\left({Y}\right) }\right}}$ $3$\displaystyle{+{b}^{2}\left[{ E\left({ {Y}^{2} }\right)- { \left{{ E\left({Y}\right) }\right} }^{2} }\right]}$

$3$\displaystyle{={a}^{2}V\left({X}\right)+2abC\left({ X,Y }\right)+{b}^{2}V\left({Y}\right)}$

∵ $3$\displaystyle{V\left({X}\right)=E\left({ {X}^{2} }\right)- {\left{{ E\left({X}\right) }\right}}^{2}}$ (ここを参照)， $3$\displaystyle{C\left({ X,Y }\right)=E\left({ XY }\right)- E\left({X}\right)E\left({Y}\right)}$ (ここを参照)

ホーム>>カテゴリー分類>>確率・統計>> $3$E\left({X}\right)$ ， $3$V\left({X}\right)$ ， $3$C\left({ X,Y }\right)$ の計算則>> $3$C\left({ X,Y }\right)=0$

最終更新日： 2026年4月8日