2
$\begingroup$

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?

$\endgroup$
1
  • 1
    $\begingroup$ Replacing $h$ by $e^y$ reduces this to a problem involving the Normal distribution. $\endgroup$
    – whuber
    Commented Jun 18, 2016 at 14:54

1 Answer 1

1
$\begingroup$

Answered in comments by whuber: Replacing $h$ by $e^y$ reduces this to a problem involving the Normal distribution.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.