Log-Normal Distribution
The Log-Normal Distribution is a statistical distribution where a random variable is said to follow a log-normal distribution if the logarithm of the variable is normally distributed. This type of distribution is often used to model positively skewed data occurring in various natural and social phenomena, such as income, city populations, stock prices, etc.
Definition: A log-normal distribution is a statistical distribution where a random variable's logarithm is normally distributed. This distribution is often used to model positively skewed data in natural and social phenomena, such as income, city populations, and stock prices.
Origin: The concept of the log-normal distribution was first introduced by British statistician Francis Galton in the late 19th century. He observed that many natural phenomena had variables whose logarithms were normally distributed, leading to the introduction of the log-normal distribution.
Categories and Characteristics: The main characteristic of a log-normal distribution is its positive skewness, meaning most data points are concentrated in a smaller range, while fewer data points are spread over a larger range. Its probability density function (PDF) is:
$$f(x; heta, eta) = rac{1}{x eta \sqrt{2\pi}} e^{- rac{(ln x - heta)^2}{2 eta^2}}$$
where ( heta) is the location parameter and (eta) is the scale parameter. The log-normal distribution is widely used in finance, economics, environmental science, and other fields.
Specific Cases:
1. Stock Price Fluctuations: In financial markets, stock price changes are often assumed to follow a log-normal distribution. This is because stock prices cannot be negative, and their logarithms tend to follow a normal distribution.
2. Income Distribution: In economics, the distribution of individual or household income typically shows positive skewness, with fewer high-income earners and most people concentrated in the lower to middle-income range. This phenomenon can be described using a log-normal distribution.
Common Questions:
1. How to determine if data follows a log-normal distribution? You can plot the histogram or QQ plot of the logarithm of the data. If the logarithm values show a normal distribution, the original data may follow a log-normal distribution.
2. What is the difference between a log-normal distribution and a normal distribution? Normal distribution data can take any real value, while log-normal distribution data can only take positive values.
