A random variable $Y$ has lognormal ($\mu,\sigma$) distribution if its probability density function is $$ f(y)=\frac{1}{y\sigma\sqrt{2\pi}}exp-\frac{(\ln y-\mu)^2}{2\sigma^2}$$ its CDF will be $$\hat F(y)=\Phi\left(\frac{\log(y)-\mu}{\sigma}\right)$$ If transformation is made as $\mu \rightarrow \log(x)$ and $\sigma \rightarrow \sqrt{4\log(1+h)}$ where $h$ is the bandwidth and $x$ is gird then what will be CDF for $$ f(y)=\frac{1}{y\sqrt{8\log(1+h)\pi}}exp[-\frac{(\ln y-\ln x)^2}{8\ln (1+h)}]$$
Either it is same as usual with transformation like $$\hat F(x)= \Phi\left(\frac{\log(y)-\log(x)}{\sqrt{4\log(1+h)}}\right)$$ or I need to again integrate it with respect to $x$ ? Please share your precious views or solution. Can someone kindly integrate 2nd $f(y)$ with respect to $x$ for limit $(0,x)$