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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)

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$$\mathrm{Prob}\{a\leq x\leq b,\ c\leq y\leq d\}=\mathrm{Prob}\{a\leq x\leq b\}\mathrm{Prob}\{c\leq y\leq d\}$$

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(1) $\Longrightarrow$ (2)

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$$\mathrm{Prob}\{a\leq x\leq b,\ c\leq y\leq d\}=\mathrm{Prob}\{a\leq x\leq b\}\mathrm{Prob}\{c\leq y\leq d\}$$

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

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

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

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

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$$F(b,d)-F(b,c)=G(b)\{H(d)-H(c)\}$$

$B$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

$$F(b,d)=G(b)H(d)$$

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(2) $\Longrightarrow$ (3)

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

(3) $\Longrightarrow$ (1)

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