Log-normal Probability Calculator

\( \)\( \)\( \)\( \)\( \)\( \) 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.
Graph of log-normal distributions

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.

\( \quad \; \mu = \)
\( \quad \; \sigma = \)
\( \quad \; x = \)
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