標本共分散の期待値

母集団の変量 $3$\text{X}$ ， $3$\text{Y}$ ，それぞれの母平均を $3$\hspace{1}{{\mu}}_{x}$ ， $3$\hspace{1}{{\mu}}_{y}$ ，母共分散を $3$\displaystyle{\hspace{1}{{\sigma}}_{ xy }}$ とする．標本として $3$n$ 個の $3$x$ と $3$y$ の組を取り出したときの標本共分散の期待値は $3$\displaystyle{\displaystyle{\frac{\hspace{2} n- 1 \hspace{2}}{\hspace{2}n\hspace{2}}\hspace{1}{{\sigma}}_{ xy }}}$ になる．

■導出

母集団の中から、 $3$x$ と $3$y$ の組の $3$n$ 個の標本を取り出す（取り出した標本は，すぐに母集団に戻し，次の標本を取り出す復元抽出）試行を考える．

1回目の試行で取り出したデータを

$3$\displaystyle{\left({ \hspace{1}{x}_{ 11 },\hspace{1}{y}_{ 11 } }\right)}$ ， $3$\displaystyle{\left({ \hspace{1}{x}_{ 12 },\hspace{1}{y}_{ 12 } }\right)}$ ， $3$\cdots$ ， $3$\displaystyle{\left({ \hspace{1}{x}_{ 1n },\hspace{1}{y}_{ 1n } }\right)}$

2回目の試行で取り出したデータを

$3$\displaystyle{\left({ \hspace{1}{x}_{ 21 },\hspace{1}{y}_{ 21 } }\right)}$ ， $3$\displaystyle{\left({ \hspace{1}{x}_{ 22 },\hspace{1}{y}_{ 22 } }\right)}$ ， $3$\cdots$ ， $3$\displaystyle{\left({ \hspace{1}{x}_{ 2n },\hspace{1}{y}_{ 2n } }\right)}$

$3$i$ 回目の試行で取り出したデータを

$3$\displaystyle{\left({ \hspace{1}{x}_{ i1 },\hspace{1}{y}_{ i1 } }\right)}$ ， $3$\displaystyle{\left({ \hspace{1}{x}_{ i2 },\hspace{1}{y}_{ i2 } }\right)}$ ， $3$\cdots$ ， $3$\displaystyle{\left({ \hspace{1}{x}_{ in },\hspace{1}{y}_{ in } }\right)}$

と表記する．

各試行( $3$i$ 回目)の試行で得られた $3$ \hspace{1}{x}_{ i1 }$ ， $3$ \hspace{1}{y}_{ i1 }$ ， $3$ \hspace{1}{x}_{ i2 }$ ， $3$ \hspace{1}{y}_{ i2 }$ ， $3$\cdots$ ， $3$\hspace{1}{x}_{ in }$ ， $3$\hspace{1}{y}_{ in }$ が生じる各確率は一定の確率で表されるので確率変数になる．それらの確率変数を $3$ \hspace{1}{X}_{ 1 }$ ， $3$ \hspace{1}{Y}_{ 1 }$ ， $3$ \hspace{1}{X}_{ 2 }$ ， $3$ \hspace{1}{Y}_{ 2 }$ ， $3$\cdots$ ， $3$\hspace{1}{X}_{ n }$ ， $3$\hspace{1}{Y}_{ n }$ とする．また，各試行の標本の変量 $3$X$ の平均を $3$\displaystyle{ \hspace{1}{{\mu}}_{ x\left({i}\right) } }$ ，変量 $3$Y$ の平均を $3$\displaystyle{ \hspace{1}{{\mu}}_{ y\left({i}\right) } }$ ，共分散を $3$\displaystyle{\hspace{1}{{\sigma}}_{ xy$i$ }}$ とする． $3$\displaystyle{ \hspace{1}{{\mu}}_{ x\left({i}\right) } }$ ， $3$\displaystyle{ \hspace{1}{{\mu}}_{ y\left({i}\right) } }$ ， $3$\displaystyle{\hspace{1}{{\sigma}}_{ xy$i$ }}$ も確率変数となり，それぞれを， $3$M$ ， $3$N$ ， $3$\hspace{1}{T}_{ XY }$ で表すことにする．以上の内容を表でまとめると以下のようになる．

試行	$3$\displaystyle{\hspace{1}{X}_{1}}$	$3$\displaystyle{\hspace{1}{Y}_{1}}$	$3$\displaystyle{\hspace{1}{X}_{2}}$	$3$\displaystyle{\hspace{1}{Y}_{2}}$	･･･	$3$\displaystyle{\hspace{1}{X}_{j}}$	$3$\displaystyle{\hspace{1}{Y}_{j}}$	･･･	$3$\displaystyle{\hspace{1}{X}_{n}}$	$3$\displaystyle{\hspace{1}{Y}_{n}}$	$3$M$	$3$N$	$3$ \hspace{1}{T}_{ XY }$
1回目	$3$\hspace{1}{x}_{11}$	$3$\hspace{1}{y}_{11}$	$3$\hspace{1}{x}_{12}$	$3$\hspace{1}{y}_{12}$	･･･	$3$\hspace{1}{x}_{ 1j }$	$3$\hspace{1}{x}_{ 1j }$	･･･	$3$\hspace{1}{x}_{ 1n }$	$3$\hspace{1}{y}_{ 1n }$	$3$\hspace{1}{{\mu}}_{ x\left({1}\right) }$	$3$\hspace{1}{{\mu}}_{ y\left({1}\right) }$	$3$\hspace{1}{{\sigma}}_{ xy\left({1}\right) }$
2回目	$3$\hspace{1}{x}_{21}$	$3$\hspace{1}{y}_{21}$	$3$\hspace{1}{x}_{22}$	$3$\hspace{1}{y}_{22}$	･･･	$3$\hspace{1}{x}_{ 2j }$	$3$\hspace{1}{x}_{ 2j }$	･･･	$3$\hspace{1}{x}_{ 2n }$	$3$\hspace{1}{y}_{ 2n }$	$3$\hspace{1}{{\mu}}_{ x\left({2}\right) }$	$3$\hspace{1}{{\mu}}_{ y\left({2}\right) }$	$3$\hspace{1}{{\sigma}}_{ xy\left({2}\right) }$
$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$
$3$i$ 回目	$3$\hspace{1}{x}_{ i1 }$	$3$\hspace{1}{y}_{ i1 }$	$3$\hspace{1}{x}_{ i2 }$	$3$\hspace{1}{y}_{ i2 }$	･･･	$3$\hspace{1}{x}_{ ij }$	$3$\hspace{1}{x}_{ ij }$	･･･	$3$\hspace{1}{x}_{ in }$	$3$\hspace{1}{y}_{ in }$	$3$\hspace{1}{{\mu}}_{ x\left({i}\right) }$	$3$\hspace{1}{{\mu}}_{ y\left({i}\right) }$	$3$\hspace{1}{{\sigma}}_{ xy\left({i}\right) }$
$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$	$3$\vdots$
期待値	$3$E\left({ \hspace{1}{X}_{1} }\right)$	$3$E\left({ \hspace{1}{Y}_{1} }\right)$	$3$E\left({ \hspace{1}{X}_{2} }\right)$	$3$E\left({ \hspace{1}{Y}_{2} }\right)$	･･･	$3$E\left({ \hspace{1}{X}_{j} }\right)$	$3$E\left({ \hspace{1}{Y}_{j} }\right)$	･･･	$3$E\left({ \hspace{1}{X}_{n} }\right)$	$3$E\left({ \hspace{1}{Y}_{n} }\right)$	$3$E\left({M}\right)$	$3$E\left({N}\right)$	$3$E\left({ \hspace{1}{T}_{ XY } }\right)$

$3$\displaystyle{\hspace{1}{{\mu}}_{ x\left({i}\right) }=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\sum }\limits_{ j=1 }^{n}\hspace{1}{x}_{ ij }}$ ･･････(1)

$3$\displaystyle{\hspace{1}{{\mu}}_{ y\left({i}\right) }=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\sum }\limits_{ j=1 }^{n}\hspace{1}{y}_{ ij }}$ ･･････(2)

$3$\displaystyle{\hspace{1}{{\sigma}}_{ xy$i$ }=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}\left({ \hspace{1}{x}_{ ij }- \hspace{1}{{\mu}}_{ x\left({i}\right) } }\right)\left({ \hspace{1}{y}_{ ij }- \hspace{1}{{\mu}}_{ y\left({i}\right) } }\right)}$ ･･････(3)

(1)，(2)，(3)を確率変数で表わすと

$3$\displaystyle{ M=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\displaystyle{{\sum }\limits_{ j=1 }^{n} \hspace{1}{X}_{j} } }$ ･･････(4)

$3$\displaystyle{ N=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\displaystyle{{\sum }\limits_{ j=1 }^{n} \hspace{1}{Y}_{j} } }$ ･･････(5)

$3$\displaystyle{\hspace{1}{T}_{ XY }=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\sum }\limits_{ j=1 }^{n}\left({ \hspace{1}{X}_{j}- M }\right)\left({ \hspace{1}{Y}_{j}- N }\right)}$ ･･････(6)

となる．

変量 $3$X$ 母平均を $3$ \hspace{1}{{\mu}}_{x}$ ，変量 $3$Y$ 母平均を $3$ \hspace{1}{{\mu}}_{y}$ ，母共分散を $3$\hspace{1}{{\sigma}}_{ xy }$ とすると，前提条件より

$3$\displaystyle{ E\left({ \hspace{1}{X}_{j} }\right)=\hspace{1}{{\mu}}_{X} }$ ･･････(7)

$3$\displaystyle{ E\left({ \hspace{1}{Y}_{j} }\right)=\hspace{1}{{\mu}}_{Y} }$ ･･････(8)

$3$\displaystyle{E\left({ \left({ \hspace{1}{X}_{j}- \hspace{1}{{\mu}}_{x} }\right)\left({ \hspace{1}{Y}_{j}- \hspace{1}{{\mu}}_{y} }\right) }\right)}$ $3$\displaystyle{=C\left({ \hspace{1}{X}_{j},\hspace{1}{Y}_{j} }\right)=\hspace{1}{{\sigma}}_{ xy }}$ ･･････(9)

である．

確率変数 $3$\hspace{1}{T}_{ XY }$ の期待値を，(7)，(8)，(9)の関係を用いて，母共分散を $3$\hspace{1}{{\sigma}}_{ xy }$ を用いて表すことを試みる．

ます，(6)の右辺の式変形をする．

$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}\left({ \hspace{1}{X}_{j}- M }\right)\left({ \hspace{1}{Y}_{j}- N }\right)}$ $3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}\left{{ \left({ \hspace{1}{X}_{j}- \hspace{1}{{\mu}}_{x} }\right)- \left({ M- \hspace{1}{{\mu}}_{x} }\right) }\right}\left{{ \left({ \hspace{1}{j}_{i}- \hspace{1}{{\mu}}_{y} }\right)- \left({ N- \hspace{1}{{\mu}}_{y} }\right) }\right}}$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}\left({ \left({ \hspace{1}{X}_{j}- \hspace{1}{{\mu}}_{x} }\right)\left({ \hspace{1}{Y}_{j}- \hspace{1}{{\mu}}_{x} }\right)- \left({ M- \hspace{1}{{\mu}}_{x} }\right)\left({ \hspace{1}{Y}_{j}- \hspace{1}{{\mu}}_{y} }\right) }$ $3$\displaystyle{\left.{ - \left({ \hspace{1}{X}_{j}- \hspace{1}{{\mu}}_{x} }\right)\left({ N- \hspace{1}{{\mu}}_{y} }\right)+\left({ M- \hspace{1}{{\mu}}_{x} }\right)\left({ N- \hspace{1}{{\mu}}_{y} }\right) }\right\)}$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}\left({ \hspace{1}{X}_{j}- \hspace{1}{{\mu}}_{x} }\right)\left({ \hspace{1}{Y}_{j}- \hspace{1}{{\mu}}_{y} }\right)}$ $3$\displaystyle{- \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}\left({ M- \hspace{1}{{\mu}}_{x} }\right)\left({ \hspace{1}{Y}_{j}- \hspace{1}{{\mu}}_{y} }\right)}$ $3$\displaystyle{- \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}\left({ \hspace{1}{X}_{j}- \hspace{1}{{\mu}}_{x} }\right)\left({ N- \hspace{1}{{\mu}}_{y} }\right)}$ $3$\displaystyle{+\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}\left({ M- \hspace{1}{{\mu}}_{x} }\right)\left({ N- \hspace{1}{{\mu}}_{y} }\right)}$

紛らわしいのを避けるため，(4)より $3$\displaystyle{ M=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\displaystyle{{\sum }\limits_{ k=1 }^{n} \hspace{1}{X}_{k} } }$ ，(5)より $3$\displaystyle{ N=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\displaystyle{{\sum }\limits_{ l=1 }^{n} \hspace{1}{Y}_{l} } }$ と考えると． $3$\displaystyle{\left({ M- \hspace{1}{{\mu}}_{x} }\right)}$ ， $3$\displaystyle{\left({ N- \hspace{1}{{\mu}}_{y} }\right)}$ は $3$\displaystyle{ {\displaystyle{\sum }}\limits_{ j=1 }^{n} }$ の外にくくり出せる．

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}\left({ \hspace{1}{X}_{j}- \hspace{1}{{\mu}}_{x} }\right)\left({ \hspace{1}{Y}_{j}- \hspace{1}{{\mu}}_{y} }\right)}$ $3$\displaystyle{- \left({ M- \hspace{1}{{\mu}}_{x} }\right)\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}\left({ \hspace{1}{Y}_{j}- \hspace{1}{{\mu}}_{y} }\right)}$ $3$\displaystyle{- \left({ N- \hspace{1}{{\mu}}_{y} }\right)\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}\left({ \hspace{1}{X}_{j}- \hspace{1}{{\mu}}_{x} }\right)}$ $3$\displaystyle{+\left({ M- \hspace{1}{{\mu}}_{x} }\right)\left({ N- \hspace{1}{{\mu}}_{y} }\right)\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}1}$

第2項，第3項，第4項を別途計算する．

$3$\displaystyle{\left({ M- \hspace{1}{{\mu}}_{x} }\right)\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}\left({ \hspace{1}{Y}_{j}- \hspace{1}{{\mu}}_{y} }\right)}$ $3$\displaystyle{=\left({ M- \hspace{1}{{\mu}}_{x} }\right)\left({ \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}\hspace{1}{Y}_{j}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}\hspace{1}{{\mu}}_{y} }\right)}$

$3$\displaystyle{=\left({ M- \hspace{1}{{\mu}}_{x} }\right)\left({ N- \hspace{1}{{\mu}}_{y} }\right)}$

$3$\displaystyle{\left({ N- \hspace{1}{{\mu}}_{y} }\right)\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}\left({ \hspace{1}{X}_{j}- \hspace{1}{{\mu}}_{x} }\right)}$ $3$\displaystyle{=\left({ N- \hspace{1}{{\mu}}_{y} }\right)\left({ \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}\hspace{1}{X}_{j}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}\hspace{1}{{\mu}}_{x} }\right)}$

$3$\displaystyle{=\left({ N- \hspace{1}{{\mu}}_{y} }\right)\left({ M- \hspace{1}{{\mu}}_{x} }\right)}$

$3$\displaystyle{\left({ M- \hspace{1}{{\mu}}_{x} }\right)\left({ N- \hspace{1}{{\mu}}_{y} }\right)\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}1}$ $3$\displaystyle{=\left({ M- \hspace{1}{{\mu}}_{x} }\right)\left({ N- \hspace{1}{{\mu}}_{y} }\right)}$

第2項を別途計算する．

$3$\displaystyle{\left({ M- \hspace{1}{{\mu}}_{x} }\right)\left({ N- \hspace{1}{{\mu}}_{y} }\right)}$ $3$\displaystyle{=\left({ \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ k=1 }^{n}\hspace{1}{X}_{k}- \hspace{1}{{\mu}}_{x} }\right)\left({ \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ l=1 }^{n}\hspace{1}{Y}_{l}- \hspace{1}{{\mu}}_{y} }\right)}$

$3$\displaystyle{=\left({ \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ k=1 }^{n}\hspace{1}{X}_{k}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ k=1 }^{n}\hspace{1}{{\mu}}_{x} }\right)\left({ \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ l=1 }^{n}\hspace{1}{Y}_{l}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ l=1 }^{n}\hspace{1}{{\mu}}_{y} }\right)}$

$3$\displaystyle{=\left({ \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ k=1 }^{n}\left({ \hspace{1}{X}_{k}- \hspace{1}{{\mu}}_{x} }\right) }\right)\left({ \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ l=1 }^{n}\left({ \hspace{1}{Y}_{l}- \hspace{1}{{\mu}}_{y} }\right) }\right)}$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} {n}^{2} \hspace{2}}\left({ {\displaystyle{\sum }}\limits_{ k=1 }^{n}\left({ \hspace{1}{X}_{k}- \hspace{1}{{\mu}}_{x} }\right) }\right)\left({ {\displaystyle{\sum }}\limits_{ l=1 }^{n}\left({ \hspace{1}{Y}_{l}- \hspace{1}{{\mu}}_{y} }\right) }\right)}$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} {n}^{2} \hspace{2}}{\sum }\limits_{ k=1 }^{n}{\sum }\limits_{ l=1 }^{n}\left({ \hspace{1}{X}_{k}- \hspace{1}{{\mu}}_{x} }\right)\left({ \hspace{1}{Y}_{l}- \hspace{1}{{\mu}}_{y} }\right)}$

この結果より

$3$\displaystyle{E\left({ \hspace{1}{T}_{ XY } }\right)=E\left({ \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\sum }\limits_{ j=1 }^{n}\left({ \hspace{1}{X}_{j}- \hspace{1}{{\mu}}_{x} }\right)\left({ \hspace{1}{Y}_{j}- \hspace{1}{{\mu}}_{y} }\right)- \frac{\hspace{2}1\hspace{2}}{\hspace{2} {n}^{2} \hspace{2}}{\sum }\limits_{ k=1 }^{n}{\sum }\limits_{ l=1 }^{n}\left({ \hspace{1}{X}_{k}- \hspace{1}{{\mu}}_{x} }\right)\left({ \hspace{1}{Y}_{l}- \hspace{1}{{\mu}}_{y} }\right) }\right)}$

$3$\displaystyle{=E\left({ \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\displaystyle{\sum }}\limits_{ j=1 }^{n}\left({ \hspace{1}{X}_{j}- \hspace{1}{{\mu}}_{x} }\right)\left({ \hspace{1}{Y}_{j}- \hspace{1}{{\mu}}_{y} }\right) }\right)}$ $3$\displaystyle{- E\left({ \frac{\hspace{2}1\hspace{2}}{\hspace{2} {n}^{2} \hspace{2}}{\sum }\limits_{ k=1 }^{n}{\sum }\limits_{ l=1 }^{n}\left({ \hspace{1}{X}_{k}- \hspace{1}{{\mu}}_{x} }\right)\left({ \hspace{1}{Y}_{l}- \hspace{1}{{\mu}}_{y} }\right) }\right)}$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\sum }\limits_{ j=1 }^{n}E\left({ \left({ \hspace{1}{X}_{j}- \hspace{1}{{\mu}}_{x} }\right)\left({ \hspace{1}{Y}_{j}- \hspace{1}{{\mu}}_{y} }\right) }\right)}$ $3$\displaystyle{- \frac{\hspace{2}1\hspace{2}}{\hspace{2} {n}^{2} \hspace{2}}{\sum }\limits_{ k=1 }^{n}{\sum }\limits_{ l=1 }^{n}E\left({ \left({ \hspace{1}{X}_{k}- \hspace{1}{{\mu}}_{x} }\right)\left({ \hspace{1}{Y}_{l}- \hspace{1}{{\mu}}_{y} }\right) }\right)}$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}{\sum }\limits_{ j=1 }^{n}C\left({ \hspace{1}{X}_{j},\hspace{1}{Y}_{j} }\right)}$ $3$\displaystyle{- \frac{\hspace{2}1\hspace{2}}{\hspace{2} {n}^{2} \hspace{2}}{\sum }\limits_{ k=1 }^{n}{\sum }\limits_{ l=1 }^{n}C\left({ \hspace{1}{X}_{k},\hspace{1}{Y}_{l} }\right)}$

$3$\displaystyle{k=l}$ のとき，(9)より

$3$\displaystyle{C\left({ \hspace{1}{X}_{k},\hspace{1}{Y}_{l} }\right)=\hspace{1}{{\sigma}}_{ xy }}$

$3$\displaystyle{k\neq l}$ のとき， $3$\displaystyle{ \hspace{1}{X}_{k} }$ と $3$\displaystyle{ \hspace{1}{Y}_{l} }$ は互いに独立である．よって

$3$\displaystyle{C\left({ \hspace{1}{X}_{k},\hspace{1}{Y}_{l} }\right)=0}$ (ここを参照)

備考： $3$\displaystyle{{\sum }\limits_{ k=1 }^{n}{\sum }\limits_{ l=1 }^{n}C\left({ \hspace{1}{X}_{k},\hspace{1}{Y}_{l} }\right)}$ において， $3$\displaystyle{k=l}$ となるのは $3$n$ 回である．

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}n\hspace{1}{{\sigma}}_{ xy }- \frac{\hspace{2}1\hspace{2}}{\hspace{2} {n}^{2} \hspace{2}}n\hspace{1}{{\sigma}}_{ xy }}$

$3$\displaystyle{=\hspace{1}{{\sigma}}_{ xy }- \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}n\hspace{1}{{\sigma}}_{ xy }}$

$3$\displaystyle{=\frac{\hspace{2} n- 1 \hspace{2}}{\hspace{2}n\hspace{2}}\hspace{1}{{\sigma}}_{ xy }}$

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最終更新日： 2026年5月6日