lognormaljakaumalla

The log-normal distribution, sometimes referred to as lognormaljakaumalla in certain contexts, is a continuous probability distribution of a random variable whose logarithm is normally distributed. This means that if a random variable X follows a log-normal distribution, then the natural logarithm of X, ln(X), follows a normal distribution. The probability density function of a log-normal distribution is given by:

f(x; mu, sigma) = (1 / (x * sigma * sqrt(2 * pi))) * exp(-(ln(x) - mu)^2 / (2 * sigma^2))

where x > 0, mu is the mean of the distribution of ln(X), and sigma is the standard deviation of the distribution of ln(X). The parameters mu and sigma are not the mean and standard deviation of X itself.

The log-normal distribution is characterized by its right-skewed shape, meaning it has a long tail extending towards higher values. It is frequently used to model phenomena that are the result of the multiplicative product of many independent random variables. Examples of its application include modeling income distribution, the size of particles in a powder, the duration of tasks in project management, and the occurrence of certain natural phenomena like the distribution of oil reserves. The underlying assumption is that these phenomena are influenced by many small, independent effects that multiply together.