# Hybrid lognormal distribution (chart) Calculator

## Calculates a table of the probability density function, or lower or upper cumulative distribution function of the hybrid lognormal distribution, and draws the chart.

 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.

 select function probability density flower cumulative distribution Pupper cumulative distribution Q parameter ρ ρ＞0 mean μ of ρx+ln(ρx) standard deviation σ of ρx+ln(ρx), σ＞0 [ initial percentile x x≧0 increment repetition ]
 $\normal Hybrid\ Lognormal\ distribution\ HybLogN(\rho x,\mu,\sigma)\\[10](1)\qquad probability\ density\\\hspace{30}f(x,\mu,\sigma)= {\large\frac{\rho}{\sqrt{2\pi}\sigma}(1+\frac{1}{\rho x})e^{-\frac{1}{2}\left(\frac{\rho x+ln(\rho x)-\mu}{\sigma}\right)^2}}\\[10](2)\qquad lower\ cumulative\ distribution\\\hspace{30}P(x,\mu,\sigma)={\large\int_{\small 0}^{\hspace{25}\small x}}f(t,\mu,\sigma)dt\\(3)\qquad upper\ cumulative\ distribution\\\hspace{30}Q(x,\mu,\sigma)={\large\int_{\small x}^{\hspace{25}\small\infty}}f(t,\mu,\sigma)dt\\(4)\ median:\ x_c={\large \frac{hyb^{-1}(\mu)}{\rho}}\\[10]\hspace{90} hyb(\rho x)=\rho x+ln(\rho x)=y\\\hspace{90} \rho x=hyb^{-1}(y)=cyb(y)\\$

Hybrid lognormal distribution (chart)
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