Lognormal Distribution Probability Calculator

Lognormal Probability Density Function:

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

Probability Density (P):

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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 commonly used to model data that are positively skewed, such as income distributions, stock prices, or particle sizes.

2. How Does the Calculator Work?

The calculator uses the lognormal probability density function:

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

Where:

\( P \) — Probability density (dimensionless)
\( x \) — Value at which to evaluate the function
\( \mu \) — Mean of the underlying normal distribution
\( \sigma \) — Standard deviation of the underlying normal distribution

Explanation: The function describes the relative likelihood for the random variable to take on a given value.

3. Importance of Lognormal Distribution

Details: Lognormal distribution is widely used in finance, environmental studies, and engineering to model variables that must be positive and have long right tails.

4. Using the Calculator

Tips: Enter the value (x > 0), mean (μ), and standard deviation (σ > 0). 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, while lognormal is skewed to the right. Lognormal variables are always positive.

Q2: What are typical applications of lognormal distribution?
A: Modeling stock prices, income distributions, particle sizes, and other phenomena where values are positive and skewed.

Q3: How do μ and σ relate to the lognormal distribution's mean?
A: The mean of the lognormal distribution is exp(μ + σ²/2), not simply μ.

Q4: Can x be zero in lognormal distribution?
A: No, x must be positive since ln(0) is undefined.

Q5: How is this related to multiplicative processes?
A: Lognormal distribution often results from multiplicative (rather than additive) processes.