Hybrid log-normal distribution
For a random variable x with a hybrid lognormal distribution, calculates the probability density function f(x), lower cumulative probability P(x), and upper cumulative probability Q(x).
The hybrid lognormal distribution is defined by the transformation hyb(ρx)=ρx+ln(ρx) following a normal distribution, where ρ is the strength parameter.
| \(\normalsize Hybrid\ Lognormal\ distribution\ HybLogN(\rho x,\mu,\sigma)\\ (1) probability\ density\\ \hspace{30px}f(x,\mu,\sigma)= {\large\frac{\rho}{\sqrt{2\pi}\sigma}(1+\frac{1}{\rho x})e^{-\frac{1}{2}\left(\frac{\rho x+ \ln(\rho x)-\mu}{\sigma}\right)^2}}\\ (2) lower\ cumulative\ distribution\\ \hspace{30px}P(x,\mu,\sigma)={\large\int_{\small 0}^{\small x}}f(t,\mu,\sigma)dt\\ (3) upper\ cumulative\ distribution\\ \hspace{30px}Q(x,\mu,\sigma)={\large\int_{\small x}^{\small\infty}}f(t,\mu,\sigma)dt\\ (4)\ median:\ x_c={\large \frac{hyb^{-1}(\mu)}{\rho}}\\ \hspace{90px} hyb(\rho x)=\rho x+ \ln(\rho x)=y\\ \hspace{90px} \rho x=hyb^{-1}(y)=cyb(y)\\ \) |
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