19.1 Probability Density Function (PDF) of the Lognormal Distribution

The Lognormal Distribution is a continuous probability distribution used to model random variables whose natural logarithm is normally distributed. This distribution is particularly useful for modeling phenomena that are inherently positive and exhibit a skewed distribution, such as income, asset prices, or biological measurements.

Parameters

The Lognormal Distribution is defined by two parameters:

• μ (mu): The mean of the natural logarithm of the random variable. This is often referred to as the location parameter.
• σ (sigma): The standard deviation of the natural logarithm of the random variable. This is often referred to as the shape parameter.

Probability Density Function (PDF) Formula

The probability density function (PDF) of the lognormal distribution is given by the following formula:

$$ f(x; \mu, \sigma) = \frac{1}{x\sigma\sqrt{2\pi}} e^{-\frac{(\ln(x) - \mu)^2}{2\sigma^2}} $$

This formula is valid for $x > 0$.

Explanation of Terms:

• $f(x)$: The probability density at a specific value $x$.
• $x$: The random variable for which you want to find the probability density. It must be a positive value ($x > 0$).
• μ: The mean of the natural logarithm of $x$ ($\text{Mean}(\ln(x))$). This parameter influences the location of the distribution's peak.
• σ: The standard deviation of the natural logarithm of $x$ ($\text{StdDev}(\ln(x))$). This parameter controls the spread and skewness of the distribution.
• $e$: Euler's number, the base of the natural logarithm, approximately equal to 2.71828.
• $\ln(x)$: The natural logarithm of $x$. This is the transformation that makes the variable normally distributed.
• $\sqrt{2\pi}$: The square root of $2\pi$, a constant derived from the normalization factor of the normal distribution.

Key Properties

• Domain: The lognormal distribution is defined only for positive values of the random variable $x$ ($x > 0$). This is because the natural logarithm of a non-positive number is undefined.
• Skewness: The lognormal distribution is always positively skewed (or right-skewed). This means the tail on the right side of the distribution is longer or fatter than the left side.
• Shape: The PDF has a single peak (unimodal). As $x$ approaches 0, the PDF approaches infinity, and as $x$ increases, the PDF approaches 0, creating a long right tail.

How Parameters Affect the Shape

• μ (Location Parameter): Increasing μ shifts the entire distribution to the right, effectively scaling up the values of the random variable. A larger μ means the mean of the underlying normal distribution is larger, leading to larger values in the lognormal distribution.
• σ (Shape Parameter): Increasing σ leads to greater skewness and a wider spread of the distribution. A larger σ makes the distribution flatter and extends the right tail further. Conversely, a smaller σ results in a distribution that is more concentrated around its peak and less skewed.

Example

Consider modeling the height of trees in a forest. While tree heights might seem normally distributed, if we consider the growth rate (which might be proportional to current height), the final height could follow a lognormal distribution.

If the natural logarithm of tree heights has a mean ($\mu$) of 3.5 (representing an average log height) and a standard deviation ($\sigma$) of 0.5 (representing variability in the growth process), we can use the lognormal PDF to calculate the probability density of a tree having a specific height.

Summary

• PDF Formula: $$ f(x; \mu, \sigma) = \frac{1}{x\sigma\sqrt{2\pi}} e^{-\frac{(\ln(x) - \mu)^2}{2\sigma^2}}, \quad \text{for } x > 0 $$
• Parameters:
  • μ (Location): Mean of $\ln(x)$
  • σ (Shape): Standard deviation of $\ln(x)$
• Domain: $x > 0$
• Skewness: Positively skewed (right-skewed)
• Shape: Unimodal with a long right tail.

Related Concepts

• Normal Distribution: The fundamental distribution of $\ln(x)$.
• Logarithmic Transformation: The process of taking the natural logarithm of the data.
• Skewness: A measure of the asymmetry of the probability distribution.

Potential Interview Questions

• What is the probability density function (PDF) formula for the lognormal distribution?
• What do the parameters $\mu$ and $\sigma$ represent in the lognormal distribution?
• Why is the lognormal distribution only defined for $x > 0$?
• How does the shape of the lognormal distribution differ from the normal distribution?
• What is the significance of the natural logarithm in the lognormal distribution?
• Describe the skewness property of the lognormal distribution.
• How does the standard deviation ($\sigma$) affect the shape of the lognormal distribution?
• What does the mean ($\mu$) parameter indicate in the context of the lognormal distribution?
• Why does the lognormal PDF have a single peak and a long right tail?
• Can you explain why the lognormal distribution is used for modeling positive-valued data?