This calculator finds the probabilities related to the log-normal distribution.

Enter the mean, standard deviation and value for the random variable and click the `Calculate`

button.

P(X = ):

P(X < ):

P(X ≤ ):

P(X > ):

P(X ≥ ):

## Log-Normal Distribution

A log-normal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. If the random variable **X** is log-normally distributed, **Y = ln(X)** has a normal distribution. It follows that if **Y** has a normal distribution, then the exponential function of **Y**, **X = exp(Y)** has a lognormal distribution. A log-normally distributed random variable can take only positive real values.

Log-normal distributions are commonly used in finance, for example, in analysing stock prices. These distributions have two distinct characteristics from the normal distribution, which make them suitable for identifying the compound return that a stock can expect to achieve over a period of time:

- Values can only be positive i.e. lower bound of zero
- Distribution is skewed to the right i.e. it has a long tail

normal distributions have zero skew and can have both negative and positive values.

The Black-Scholes Options Pricing model assumes that stock prices follow a log-normal distribution, based on the principle that asset prices have a lower bound of zero.