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Next: $B$3$NJ8=q$K$D$$$F(B... (PDF ե: independ.pdf)

$BJ?@.(B 13 $BG/(B 6 $B7n(B 08 $BF|(B
$BO"B3J,I[$N3NN(JQ?t$NFHN)@->r7o(B
$B?73c9)2JBg3X(B $B>pJsEE;R9)3X2J(B $BC]LnLP<#(B

$BO"B3J,I[$N>l9g!"
$BDjM}(B 1

  1. $x$ $B$H(B $y$ $B$OFHN)(B
  2. $F(x,y)=G(x)H(y)$ ( $F(x,y),G(x),H(y)$: $B$=$l$>$l(B $(x,y), x, y$ $B$NJ,I[4X?t(B)
  3. $f(x,y)=g(x)h(y)$ ( $f(x,y),g(x),h(y)$: $B$=$l$>$l(B $(x,y), x, y$ $B$NL)EY4X?t(B)


$B$J$*!"(B(1) $B$NDj5A$O
$BDj5A(B 2

$BO"B3J,I[$K=>$&(B $x$ $B$H(B $y$ $B$,(B $BFHN)(B $B$G$"$k$H$O!"(B $BG$0U$N, $c\leq d$ $B$KBP$7$F(B

\begin{displaymath}
\mathrm{Prob}\{a\leq x\leq b,\ c\leq y\leq d\}=\mathrm{Prob}\{a\leq x\leq b\}\mathrm{Prob}\{c\leq y\leq d\}
\end{displaymath}

$B$H$J$k$3$H!#(B


$B>ZL@(B

(1) $\Longrightarrow$ (2)

$BG$0U$N(B $a\leq b$, $c\leq d$ $B$KBP$7$F(B

\begin{displaymath}
\mathrm{Prob}\{a\leq x\leq b,\ c\leq y\leq d\}=\mathrm{Prob}\{a\leq x\leq b\}\mathrm{Prob}\{c\leq y\leq d\}
\end{displaymath}

$B$G$"$j!"0lJ}!"(B

\begin{eqnarray*}
&& \mathrm{Prob}\{a\leq x\leq b,\ c\leq y\leq d\}=F(b,d)-F(a,...
...q b\}=G(b)-G(a)\\
&& \mathrm{Prob}\{c\leq y\leq d\}=H(d)-H(c)
\end{eqnarray*}



$B$J$N$G(B

\begin{displaymath}
F(b,d)-F(a,d)-F(b,c)+F(a,c)=\{G(b)-G(a)\}\{H(d)-H(c)\}
\end{displaymath}

$B$3$3$G(B $a\rightarrow -\infty$ $B$H$9$k$H(B

\begin{eqnarray*}
&& G(a)=\mathrm{Prob}\{x\leq a\}\rightarrow 0,\\
&& F(a,d)=...
...ob}\{x\leq a\}\rightarrow 0
\mbox{ $B$h$j(B } F(a,c)\rightarrow 0
\end{eqnarray*}



$B$H$J$k$N$G(B

\begin{displaymath}
F(b,d)-F(b,c)=G(b)\{H(d)-H(c)\}
\end{displaymath}

$B$H$J$k!#F1MM$K(B $c\rightarrow -\infty$ $B$H$9$k$H(B $H(c)\rightarrow 0$, $F(b,c)\leq H(c)$ $B$h$j(B $F(b,c)\rightarrow 0$ $B$H$J$j!"7k6I(B

\begin{displaymath}
F(b,d)=G(b)H(d)
\end{displaymath}

$B$H$J$k!#(B$b$, $d$ $B$OG$0U$J$N$G(B (2) $B$,$$$($?$3$H$K$J$k!#(B

(2) $\Longrightarrow$ (3)


\begin{displaymath}
f(x,y)=F_{xy}(x,y)=\{G(x)H(y)\}_{xy}
=\{G'(x)H(y)\}_{y} = G'(x)H'(y)=g(x)h(y)
\end{displaymath}

(3) $\Longrightarrow$ (1)

\begin{eqnarray*}
\lefteqn{\mathrm{Prob}\{a\leq x\leq b,\ c\leq y\leq d\}}\\
...
... & \mathrm{Prob}\{a\leq x\leq b\}\mathrm{Prob}\{c\leq y\leq d\}
\end{eqnarray*}







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Next: $B$3$NJ8=q$K$D$$$F(B...
Shigeharu TAKENO
2001$BG/(B 8$B7n(B 9$BF|(B