# 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.

## 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.

 $$\quad \; \mu =$$ 3 $$\quad \; \sigma =$$ 2 $$\quad \; x =$$ 2 Decimal Places Desired = 5