Lognormal Moments and Scale

The lognormal distribution models a random variable X whose logarithm is normally distributed. If ln X ~ Normal(μ, σ²), then X only takes positive values. The parameters μ and σ are the mean and standard deviation of the variable's natural logarithm, not of X itself.

Moments and Characteristics

For a continuous lognormal variable, the key characteristics are derived from its moment generating function (MGF):

• Mean: E[X] = exp(μ + σ²/2)
• Median: exp(μ)
• Mode: exp(μ - σ²)
• Variance: (exp(σ²) - 1)exp(2μ + σ²)

The NIST handbook provides standard parameterizations for lognormal variables along with PDF/CDF definitions and properties. The point-mass behavior at sigma=0 is explicitly extended in this module as noted in the contract.

Because σ² > 0 for any continuous lognormal distribution, the mode is always strictly less than the median, which is strictly less than the mean. This indicates a positive (right) skew.

Degenerate Case: Point Mass

When σ = 0, the variance of ln X is zero, meaning ln X takes the constant value μ with probability 1. Therefore, X = exp(μ) with probability 1. This degenerate case is an explicit point mass at exp(μ). For a point mass, the ordinary continuous probability density is unavailable (or represented as a Dirac delta function), while the cumulative distribution function (CDF) is a step function that jumps from 0 to 1 at exp(μ). The mean, median, and mode all equal exp(μ), and the variance is 0.

Density and CDF

The probability density function (PDF) for x > 0 is:

f(x) = (1 / (x * σ * √(2π))) * exp( -(ln(x) - μ)² / (2σ²) )

The cumulative distribution function (CDF) is given by:

F(x) = Φ( (ln(x) - μ) / σ )

where Φ is the standard normal CDF.

Experiment

1. Set Log mean to 0 and Log standard deviation to 1. Observe that the mean (1.648...) is greater than the median (1.0), which is greater than the mode (0.367...).
2. Change Probe x to 1. The CDF is exactly 0.5, since the median is 1.
3. Set Log standard deviation to 0. The distribution becomes a Point mass at exp(Log mean). The density becomes unavailable, and the CDF becomes a step function.
4. Set Probe x to 0.5 when Log standard deviation is 0. Observe the CDF is 0. Change Probe x to 1, and the CDF jumps to 1.
5. Change Log mean to -1 and Log standard deviation to 1.5. Notice how the positive skew becomes even more pronounced.

Explore the interactive module here: https://riano.app/module/lognormal-moments

Limitations

The plotted finite interval represents only a portion of the support (0, ∞). The lognormal distribution has an infinite positive tail which is not fully visible on the finite plot. Extreme tail probabilities should be calculated accurately to avoid catastrophic cancellation in floating-point math.

References

• NIST lognormal distribution handbook

The NIST handbook provides standard parameterizations for lognormal variables along with PDF/CDF definitions and properties.