# Hybrid lognormal distribution (percentile) Calculator

## Calculates the percentile from the lower or upper cumulative distribution function of the hybrid lognormal distribution.

 Hybrid lognormal distribution is the distribution function changed the percentile "x" to "ln(x)+x " in the normal distribution function. It is also calculated by the random variable hyb(ρx)=ρx+ln(ρx) including the parameter ρ.
cumulative mode
 lower P upper Q
cumulative distribution
 0≦P,Q≦1
parameter ρ
 ρ＞0
mean μ
 of ρx+ln(ρx)
standard deviation σ
 of ρx+ln(ρx), σ＞0
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 $\normal Hybrid\ Lognormal\ distribution\ HybLogN(\rho x,\mu,\sigma)\\[10](1)\qquad probability\ density\\\hspace{30}f(x,\mu,\sigma)= {\large\frac{\rho}{\sqrt{2\pi}\sigma}(1+\frac{1}{\rho x})e^{-\frac{1}{2}\left(\frac{\rho x+ln(\rho x)-\mu}{\sigma}\right)^2}}\\[10](2)\qquad lower\ cumulative\ distribution\\\hspace{30}P(x,\mu,\sigma)={\large\int_{\small 0}^{\hspace{25}\small x}}f(t,\mu,\sigma)dt\\(3)\qquad upper\ cumulative\ distribution\\\hspace{30}Q(x,\mu,\sigma)={\large\int_{\small x}^{\hspace{25}\small\infty}}f(t,\mu,\sigma)dt\\(4)\ median:\ x_c={\large \frac{hyb^{-1}(\mu)}{\rho}}\\[10]\hspace{90} hyb(\rho x)=\rho x+ln(\rho x)=y\\\hspace{90} \rho x=hyb^{-1}(y)=cyb(y)\\$

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