91Ó°ÊÓ

A and b are close together then

\(\int\limits_a^b {r\left( t \right)} dt \approx \left( {b - a} \right)r\left( {\frac{{a + b}}{2}} \right)\)

X and y have the joint pdf f

The conditional pdf of\({g_1}\left( {x,y} \right)\):

\({g_1}\left( {x,y} \right) = \mathop {\lim }\limits_{\varepsilon \to 0} \frac{\partial }{{\partial x}}{\rm P}\left( {X \le x|y - \varepsilon < Y \le y + \varepsilon } \right)\)

\({f_2}\)be the marginal pdf of Y.

Hence we approximate

\({\rm P}\left( {y - \varepsilon < Y \le y + \varepsilon } \right) = \int\limits_{y - \varepsilon }^{y + \varepsilon } {{f_2}\left( t \right)dt} .\)

\( \approx 2\varepsilon {f_2}\left( y \right)\)

Step 3: When\({\rm P}\left( {X \le x\,and\,y - \varepsilon < Y \le y + \varepsilon } \right)\)and\( = \int\limits_{ - \infty }^x {\int\limits_{y - \varepsilon }^{y + \varepsilon } {f\left( {s,t} \right)dtds} } \)

For each s, we approximate

\(\int\limits_{y - \varepsilon }^{y + \varepsilon } {f\left( {s,t} \right)dt \approx 2\varepsilon f\left( {s,y} \right)} .\)

\(\int\limits_a^b {r\left( t \right)} dt \approx \left( {b - a} \right)r\left( {\frac{{a + b}}{2}} \right)\)with\(r\left( t \right) = f\left( {s,t} \right)\)

Therefore

\({\rm P}\left( {X \le x\,and\,y - \varepsilon < Y \le y + \varepsilon } \right) = \int\limits_{ - \infty }^x {\int\limits_{y - \varepsilon }^{y + \varepsilon } {f\left( {s,t} \right)dtds} } \)

\( \approx 2\varepsilon \int\limits_{ - \infty }^\infty {f\left( {s,y} \right)ds} \)

Taking the ratio of the approximation in part(b) to the approximation in part (a)

We get,

\(\)\(\begin{aligned}{}{\rm P}\left( {X \le {\raise0.7ex\hbox{$x$} \!\mathord{\left/{\vphantom {x {y - \varepsilon}}}\right.}\!\lower0.7ex\hbox{${y - \varepsilon }$}}\, < Y \le y + \varepsilon } \right) &= \frac{{{\rm P}\left( {X \le x,y - \varepsilon \, < Y \le y + \varepsilon } \right)}}{{{\rm P}\left( {y - \varepsilon \, < Y \le \varepsilon } \right)}}\\ \approx \frac{{\int\limits_{ - \infty }^x {f\left( {s,y} \right)ds} }}{{{f_2}\left( y \right)}}\end{aligned}\)

So, \(P = \int\limits_{ - \infty }^x {{g_1}\left( {s|y} \right)} ds.\)