\( \)\( \)\( \)\( \)\( \)\( \) An easy to use calculator to compute the cumulative probability distribution of the log-normal distribution whose probability density function is defined below.
The cumulative probability \( F_X(a) \) of the log-normal distribution may be expressed by
\[ F_X(a) = \dfrac{1}{2} \left(1+\text{Erf} \left( \dfrac{\ln a - \mu}{\sigma \sqrt{2}} \right) \right) \]
where \( \text{Erf}(x) \) is the error function.
1) The mean is given by
\( \qquad e^{(\mu + \frac{\sigma^2}{2})}\)
2) The median is given by
\( \qquad e^{\mu} \)
3) The mode is given by
\( \qquad e^{\mu - \sigma^2} \)
4) The variance is given by
\( \qquad (e^{\sigma^2} - 1)(e^{2\mu+\sigma^2}) \)
5) The standard deviation is given by
\( \qquad \sqrt {(e^{\sigma^2} - 1)(e^{2\mu+\sigma^2})} \)
Answer