What is log-normal distribution?

The log-normal distribution is a probability distribution of a random variable whose logarithm is normally distributed.

The log-normal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. It is often used to model variables that are positive and skewed, such as income, stock prices, and population sizes. The distribution is characterized by two parameters: the mean and standard deviation of the logarithm of the variable.

The probability density function (PDF) of the log-normal distribution is given by:

f(x) = (1 / (x * σ * √(2π))) * e^(-(ln(x) - μ)^2 / (2σ^2))

where x is the value of the random variable, μ is the mean of the logarithm of the variable, σ is the standard deviation of the logarithm of the variable, and e is the base of the natural logarithm.

The cumulative distribution function (CDF) of the log-normal distribution is given by:

F(x) = Φ((ln(x) - μ) / σ)

where Φ is the standard normal cumulative distribution function.

The expected value of the log-normal distribution is:

E(X) = e^(μ + σ^2 / 2)

and the variance is:

Var(X) = (e^(σ^2) - 1) * e^(2μ + σ^2)

The log-normal distribution is often used in finance and economics to model asset prices and returns, as well as in biology to model population sizes and growth rates.