Lognormalfordeling

Lognormalfordeling is a continuous probability distribution of a random variable whose natural logarithm is normally distributed. This means that if a random variable X follows a lognormal distribution, then the natural logarithm of X, denoted as ln(X), follows a normal distribution. The probability density function (PDF) of a lognormal distribution is given by f(x; mu, sigma) = (1 / (x * sigma * sqrt(2 * pi))) * exp(-((ln(x) - mu)^2) / (2 * sigma^2)) for x > 0, where mu is the mean of the underlying normal distribution of ln(X) and sigma is the standard deviation of the underlying normal distribution of ln(X).

The lognormal distribution is characterized by its positive skewness. This means that the tail of the distribution on the right side is longer than the tail on the left side. The shape of the lognormal distribution is determined by the parameters mu and sigma. As sigma increases, the distribution becomes more spread out and more skewed. The mean of a lognormal distribution is exp(mu + (sigma^2)/2), and the variance is (exp(sigma^2) - 1) * exp(2*mu + sigma^2).

The lognormal distribution is frequently used to model phenomena that are the result of multiplicative processes. Examples include the distribution of incomes, the size of particles in a sample, or the time to failure of mechanical components. It is important to note that the lognormal distribution is defined only for positive values of the random variable.