Hybrid lognormal distribution Calculator

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Calculates the probability density function and lower and upper cumulative distribution functions 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.

\(\normalsize Hybrid\ Lognormal\ distribution\ HybLogN(\rho x,\mu,\sigma)\\
(1) probability\ density\\
\hspace{30px}f(x,\mu,\sigma)= {\large\frac{\rho}{\sqrt{2\pi}\sigma}(1+\frac{1}{\rho x})e^{-\frac{1}{2}\left(\frac{\rho x+ \ln(\rho x)-\mu}{\sigma}\right)^2}}\\
(2) lower\ cumulative\ distribution\\
\hspace{30px}P(x,\mu,\sigma)={\large\int_{\small 0}^{\small x}}f(t,\mu,\sigma)dt\\
(3) upper\ cumulative\ distribution\\
\hspace{30px}Q(x,\mu,\sigma)={\large\int_{\small x}^{\small\infty}}f(t,\mu,\sigma)dt\\
(4)\ median:\ x_c={\large \frac{hyb^{-1}(\mu)}{\rho}}\\
\hspace{90px} hyb(\rho x)=\rho x+ \ln(\rho x)=y\\
\hspace{90px} \rho x=hyb^{-1}(y)=cyb(y)\\
\)

Hybrid lognormal distribution Calculator

Calculates the probability density function and lower and upper cumulative distribution functions of the hybrid lognormal distribution.

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Hybrid lognormal distribution Calculator

Calculates the probability density function and lower and upper cumulative distribution functions of the hybrid lognormal distribution.

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