Lognormal Distribution: Definition, Parameters, Functions

A probability distribution of events is normally distributed, which means that it forms a symmetrical bell-shaped curve. Logarithmic mathematics can be used to create a log-normal distribution from a normal distribution. A lognormal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Only positive real values are accepted by a random variable with a lognormal distribution. Most natural phenomena, such as tissue size, blood pressure, wealth distribution, and even the length of a chess game, demonstrate phenomena whose relative growth is independent of their size, which is true for most natural phenomena.

Key Terms: Probability, Normal Distribution, Continuous Distribution, Density Function, Logarithms, Variable, Mean, Median, Mode, Variance

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Definition of Lognormal Distribution

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The probability distribution of random variables is represented as a log-normal distribution.

If Y = ln(X) is normally distributed and X is a random variable, then X is said to be distributed lognormally. If Y has a normal distribution, then Y's exponential function will have a lognormal distribution, i.e., X = exp (Y). A lognormal process is a statistical consequence of the multiplicative product of many independent random variables, each of which is positive. Positive values across the board, skewed distributions with low mean values and high variation. Because log(x) exists exclusively for positive values.

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Probability Density Function 

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A random variable X is log-normally distributed if and only if its logarithm is normally distributed. Furthermore, the value of X must be positive.

The lognormal distribution has parameters and is a two-parameter distribution. As an example, the probability density function may be written as:

\(f\left(t^{\prime}\right)=\frac{1}{\left(\sigma^{\prime} \sqrt{2 \pi}\right)} e^{-\frac{1}{2}\left(\frac{t-\mu}{\sigma}\right)^{2}}\)

t = The time-to-failure is represented

Mean (µ) = Natural logarithms of the time-to-failure

Standard Deviation (σ) = The natural logarithms of the time-to-failure

The lognormal probability distribution may be produced by recognising that incremental areas should be identical for equal probabilities under the normal and lognormal probability distributions.

So,

\(f\left(t^{\prime}\right)=\frac{1}{\left(\sigma^{\prime} \sqrt{2 \pi}\right)} e^{-\frac{1}{2}\left(\frac{t-\mu}{\sigma}\right)^{2}}\)

Applying the derivative 

dt' = (dt)/t

From the two equations above, 

f(t) = f(t')/t

\(f\left(t \right)=\frac{1}{\left(t.\sigma^{\prime} \sqrt{2 \pi}\right)} e^{-\frac{1}{2}\left(\frac{ln(t)-\mu}{\sigma}\right)^{2}}\)

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Lognormal Distribution Curve

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Lognormal Distribution Curve

The right-hand side of the lognormal distribution curve is skewed. The probability distribution function's value starts at zero, increases, and finally decreases. The degree of skewness will rise as increases for a given. If it increases, the skewness of the probability distribution function will likewise grow.

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Parameters of Lognormal Distribution

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The shape of a log-normal distribution is determined by the following lognormal distribution parameters:

1. Shape Parameter- The standard deviation of a log normal distribution has an impact on the overall shape of the distribution. The shape parameter does not influence the graph's placement or height; rather, it changes the graph's form.
2. Scale Parameter- It is a median that causes the graph to shrink or stretch.
3. Location Parameter- The graph's location on the x-axis is determined by this parameter.

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Lognormal Distribution Functions

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The major lognormal distribution functions include mean, mode, median and variance. 

1. Mean
   The mean of the lognormal distribution is-
   m = e^{μ +σ²/2}
   Here, the mean can also be calculated-
   μ = Inm - ½ σ²
2. Mode
   The mode of the lognormal distribution is-
   Mode X = e^{μ - σ²}
   Setting the probability distribution function to 0 gives us this function, as the mode reflects the distribution's global maximum.
3. Median
   The median of the lognormal distribution is-
   Med X = e^μ
   Setting the cumulative frequency to 0.5 and calculating the resultant equation yields this result.
4. Variance
   The variance of the lognormal distribution is-
   Var X = (e^σ² - 1)e^{2μ + σ²}
   This may alternatively be expressed as (e^σ² - 1), where σ is the mean of the previous distribution.

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Applications of Lognormal Distribution

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• Solved Rubik's Cubes, whether in general or by a single person, appear to follow a lognormal distribution.
• A lognormal distribution describes how much time a person spends on internet content (jokes, news, etc.).
• In economics, to examine the income of the population other than those with higher incomes. It is also useful in analyzing the changes in the stock market.
• The duration of comments submitted in discussion forums on social media websites follows a lognormal distribution.

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Lognormal Distribution Tables

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Things to Remember

• The logarithm of a variable has a normal distribution in a continuous distribution. With S=1 and M=0, it is a generic instance of Gibrat's distribution, to which the log normal distribution reduces.
• Descriptions of fatigue failure, failure rates, and other phenomena involving a vast range of data are common applications of the lognormal distribution.
• The lognormal distribution is a two-parameter distribution with mean and standard deviation as its parameters. The mean, median, mode, and variance are the four major lognormal distribution functions.
• The lognormal distribution curve is skewed towards the right and this form is reliant on three criteria of shape, location, and scale.

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