Lognormal Distribution Calculator

Lognormal Probability Density Function (PDF):

\[ f(x) = \frac{1}{x \sigma \sqrt{2\pi}} \exp\left( -\frac{(\ln x - \mu)^2}{2\sigma^2} \right) \]

Probability Density (PDF):

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1. What is Lognormal Distribution?

The lognormal distribution is a probability distribution of a random variable whose logarithm is normally distributed. It's used when the quantity of interest must be positive and has a right-skewed distribution.

2. How Does the Calculator Work?

The calculator uses the lognormal probability density function (PDF):

\[ f(x) = \frac{1}{x \sigma \sqrt{2\pi}} \exp\left( -\frac{(\ln x - \mu)^2}{2\sigma^2} \right) \]

Where:

\( x \) — Value at which to evaluate the PDF (must be positive)
\( \mu \) — Mean of the natural logarithm of the variable
\( \sigma \) — Standard deviation of the natural logarithm of the variable (must be positive)

Explanation: The lognormal distribution is characterized by parameters μ and σ, which are the mean and standard deviation of the variable's natural logarithm.

3. Applications of Lognormal Distribution

Details: Lognormal distribution is commonly used in finance (stock prices), biology (species abundance), engineering (particle sizes), and other fields where values are positively skewed and cannot be negative.

4. Using the Calculator

Tips: Enter a positive value for x, any value for μ, and a positive value for σ. The calculator will return the probability density at point x.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between normal and lognormal distribution?
A: Normal distribution is symmetric and can have negative values, while lognormal is skewed right and only has positive values.

Q2: How are μ and σ related to the mean and variance of the distribution?
A: The mean is exp(μ + σ²/2) and variance is [exp(σ²)-1]exp(2μ + σ²).

Q3: When should I use lognormal distribution?
A: When your data is positively skewed, always positive, and the logarithm of the data appears normally distributed.

Q4: Can the lognormal PDF be greater than 1?
A: Yes, PDF values can exceed 1 since they're probability densities, not probabilities.

Q5: How do I estimate μ and σ from data?
A: Take the natural log of all data points, then calculate the mean and standard deviation of these logged values.