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The problem comes from the personal research

$h \sim \log N(\mu,\sigma^2)$ then $$f_H(h) = \frac{1}{h\sigma\sqrt{2\pi}}\exp\left[-\frac{1}{2}\left(\frac{\log h-\mu}{\sigma}\right)^2\right].$$

Here is a integration

$$P_l=\int_{-w}^{w} \int_{\frac{\ x_h}{\ y_h}(x_b-w)+l}^{\frac{\ x_h}{\ y_h}(x_b+w)+l} P_r\left( h\geqq\frac{\ z_h(y_b-l)}{y_h} \right){}dy_bdx_b $$

in this integration $$x_h,y_h,z_h,w,l $$

are constants. how to figure it out? My question is we can turn this into $$P_l=\int_{-w}^{w} \int_{\frac{\ x_h}{\ y_h}(x_b-w)+l}^{\frac{\ x_h}{\ y_h}(x_b+w)+l} \int_{\frac{\ z_h(y_b-l)}{y_h}}^{+\infty} \space { f_H(h) } \space dy_bdx_b .$$

But how to integrate the lognormal distribution?

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    $\begingroup$ Replacing $h$ by $e^y$ reduces this to a problem involving the Normal distribution. $\endgroup$ – whuber Jun 18 '16 at 14:54
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Answered in comments by whuber: Replacing $h$ by $e^y$ reduces this to a problem involving the Normal distribution.

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