How to Calculate Log Normal Distribution: Step-by-Step Guide

Introduction to Log Normal Distribution

The log normal distribution is a probability distribution of a random variable whose logarithm is normally distributed. It is commonly used in finance, engineering, and other fields to model skewed distributions. In this guide, we will walk you through the step-by-step process of calculating the log normal distribution manually.

What is the Formula for Log Normal Distribution?

The probability density function (PDF) of the log normal distribution is given by:

f(x | μ, σ) = (1 / (x * σ * √(2 * π))) * exp(-((ln(x) - μ)^2) / (2 * σ^2))

where:

• x is the value of the random variable
• μ is the mean of the logarithm of the random variable
• σ is the standard deviation of the logarithm of the random variable

Step-by-Step Calculation

To calculate the log normal distribution, follow these steps:

Step 1: Identify the Given Values

Identify the values of x, μ, and σ. These values are necessary to calculate the log normal distribution.

Step 2: Calculate the Natural Logarithm of x

Calculate the natural logarithm of x (ln(x)). This value will be used in the formula.

Step 3: Calculate the Exponent

Calculate the exponent of the formula: -((ln(x) - μ)^2) / (2 * σ^2).

Step 4: Calculate the Coefficient

Calculate the coefficient of the formula: 1 / (x * σ * √(2 * π)).

Step 5: Calculate the Log Normal Distribution

Multiply the coefficient and the exponent to get the final result.

Worked Example

Let's calculate the log normal distribution for x = 10, μ = 2, and σ = 0.5.

• Calculate the natural logarithm of x: ln(10) = 2.3026
• Calculate the exponent: -((2.3026 - 2)^2) / (2 * 0.5^2) = -((0.3026)^2) / (2 * 0.25) = -0.0918 / 0.5 = -0.1836
• Calculate the coefficient: 1 / (10 * 0.5 * √(2 * π)) = 1 / (5 * √(2 * π)) = 1 / (5 * 2.5066) = 1 / 12.533 = 0.0798
• Calculate the log normal distribution: 0.0798 * exp(-0.1836) = 0.0798 * 0.8313 = 0.0664

Common Mistakes to Avoid

When calculating the log normal distribution, make sure to:

• Use the correct values for x, μ, and σ
• Calculate the natural logarithm of x correctly
• Use the correct formula for the exponent and coefficient

When to Use a Calculator

While it is possible to calculate the log normal distribution manually, it can be time-consuming and prone to errors. In most cases, it is recommended to use a calculator or software to perform the calculation. This is especially true when dealing with large datasets or complex calculations.

Conclusion

In this guide, we have walked you through the step-by-step process of calculating the log normal distribution manually. We have also provided a worked example and highlighted common mistakes to avoid. While manual calculation is possible, it is often more convenient to use a calculator or software to perform the calculation.