# Log-normal distribution

Log-normal distribution |
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**Log-normal distribution** is a type of probability distribution where the logarithm of a random variable is normally distributed. It is often used to model data which exhibit large variations, such as stock prices, salaries, and income.

## Example of Log-normal distribution

Log-normal distributions can be seen in many real-world phenomena. Examples include the size of grains of sand, the concentration of pollutants in the air, and the size of buildings in a city. All of these phenomena display large variations, and so can be modeled using a log-normal distribution.

## Formula of Log-normal distribution

Formula of Log-normal distribution as well as its cumulative distribution function are used to describe the probability of the occurrence of different values of the random variable. Log-normal distributions are often used to model phenomena which exhibit large variations, such as stock prices, salaries, and income.

The probability density function for a log-normal distribution is given by\[f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(ln(x)-\mu)^2}{2\sigma^2}\right]}\]

where μis the mean of the logarithm of the random variable, and σ is the standard deviation of the logarithm. The cumulative distribution function is given by\[F(x)=\int_{-\infty}^{ln(x)}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(ln(x)-\mu)^2}{2\sigma^2}\right]}dx\]

## When to use Log-normal distribution

Log-normal distributions are used in many fields and applications. It is commonly used in finance, economics, and engineering to model random variables that are skewed to the right (positively skewed). Examples of when to use a log-normal distribution include:

**Stock prices**: When modeling stock prices, a log-normal distribution can be used to capture the large variations seen in stock prices.**Salaries**: When modeling salaries, a log-normal distribution can be used to capture the large variations seen in salaries.**Income**: When modeling income, a log-normal distribution can be used to capture the large variations seen in income.

## Types of Log-normal distribution

Log-normal distributions can be divided into three types, depending on the distribution of the mean and standard deviation of the logarithm of the random variable:

**Unimodal**: In this type, the mean and standard deviation of the logarithm of the random variable are both constant.**Bimodal**: In this type, the mean of the logarithm of the random variable is constant, but the standard deviation is not.**Multimodal**: In this type, the mean and standard deviation of the logarithm of the random variable are both variable.

## Steps of Log-normal distribution

Log-normal distribution involves three steps:

**Calculating the mean**: The first step is to calculate the mean of the logarithm of the random variable. This is done using the formula\[\mu = \frac{1}{n}\sum_{i=1}^{n}ln(x_i)\]

where x_{i} is the ith observation in the data set.

**Calculating the standard deviation**: The second step is to calculate the standard deviation of the logarithm of the random variable. This is done using the formula\[\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(ln(x_i)-\mu\right)^2}\]**Calculating the probability density function and cumulative distribution function**: The third step is to calculate the probability density function and the cumulative distribution function. The probability density function is given by\[f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(ln(x)-\mu)^2}{2\sigma^2}\right]}\]

and the cumulative distribution function is given by\[F(x)=\int_{-\infty}^{ln(x)}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(ln(x)-\mu)^2}{2\sigma^2}\right]}dx\]

In conclusion, log-normal distribution involves three steps: calculating the mean, calculating the standard deviation, and calculating the probability density function and cumulative distribution function.

## Advantages of Log-normal distribution

Log-normal distribution has several advantages compared to other probability distributions. Firstly, it is relatively easy to calculate and interpret. Secondly, it is a flexible distribution which can be used to model phenomena which exhibit large variations. Additionally, it can be used to calculate the probability of rare events. Finally, it can also be used to model non-linear relationships.

## Limitations of Log-normal distribution

Log-normal distributions have a few limitations. First, the distribution only works for positive values since the logarithm of a negative value is undefined. Second, the distribution is not well-suited for modeling data that are not skewed, as the log-normal distribution is always skewed. Third, the distribution assumes that the underlying data is normally distributed, which may not always be the case. Finally, the parameters of the distribution (i.e. μ and σ) are not always easy to estimate.

Log-normal distributions are often used in conjunction with other approaches to model data which exhibit large variations. Some of these approaches include:

**Monte Carlo simulation**: Monte Carlo simulation is a method of using random sampling to solve a problem. It can be used to generate random samples from a log-normal distribution, which can then be used to model the data.**Bayesian inference**: Bayesian inference is a method of using prior information to update beliefs about an unknown quantity. It can be used to update beliefs about the parameters of a log-normal distribution, such as the mean and standard deviation.**Maximum likelihood estimation**: Maximum likelihood estimation is a method of using a set of data points to estimate the parameters of a model. It can be used to estimate the parameters of a log-normal distribution from a set of data points.

In conclusion, log-normal distributions are often used in conjunction with other approaches to model data which exhibit large variations, such as Monte Carlo simulation, Bayesian inference, and maximum likelihood estimation.

## Suggested literature

- Mitchell, R. L. (1968).
*Permanence of the log-normal distribution*. JOSA, 58(9), 1267-1272. - Olsson, U. (2005).
*Confidence intervals for the mean of a log-normal distribution*. Journal of Statistics Education, 13(1).