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An easy to use calculator to compute the cumulative probability distribution of the log-normal distribution whose probability density function is defined below.

Log-Normal Distribution

The log-normal distribution is defined by [1]
\[ \displaystyle f (x) = \dfrac{1}{x \sigma \sqrt{2\pi}} \; e^{-\dfrac{(\ln x - \mu)^2}{2 {\sigma}^2}} \]
is presented.
The graph \( f(x) \) for different values of the parameters \( \mu \) and \( \sigma \) are shown below.

Cumulative Probability of Log-Normal Distribution

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.

Formulas of the Mean, Median, Mode, variance, standard deviation and Skewness of Log-normal Distribution

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
\( \qquad \sqrt {(e^{\sigma^2} - 1)(e^{2\mu+\sigma^2})} \)

Use Log-normal Distribution Probability Calculator

Enter the parameters \( \mu \) as a real number and \( \sigma \) as a positive real number.
Enter \( x \) as a positive real number.
The outputs are: the cumulative probability \( P(X \le x) = F_X(a) \), the mean, median, mode, variance and standard deviation (STDEV) defined above.

Answer

More References and Links

Statistics - James McClave et Terry Sincich - 13th Edition - 2016 - ISBN-10 : 0134080211