Log Normal Distribution

A Log-Normal Distribution models a random variable whose logarithm is normally distributed.

That is:

In simple terms:

If taking the log of your data makes it look Normal,
then your original data follow a Log-Normal Distribution.

Probability Density Function (PDF)

Where:

• μ = mean of the log-transformed variable (ln⁡ X)
• σ = standard deviation of the log-transformed variable

The total area under the curve = 1
Only defined for x > 0

Intuition

• The Normal distribution is symmetric → works well for additive effects.
• The Log-Normal distribution is skewed → models multiplicative effects (e.g., product of random factors).

If you multiply random variables, the result tends to be log-normal.
If you add random variables, the result tends to be normal.

Examples

• Income distribution - Most people earn near average, few earn much more, skewed right, all values > 0
• Stock prices - Compound percentage growth, price changes multiply over time
• Reaction times - Most are short, few long, positive and skewed
• Word frequencies - Few words are common, many are rare, power-law-like behavior

• Left plot (PDF):
• Skewed to the right (long tail)
• High probability near smaller x values, declining rapidly for larger x
• Right plot (CDF):
• Increases slowly at first, then approaches 1 as x grows

The shape is positively skewed
Defined only for x > 0

Effect of σ (Shape Parameter)

Smaller σ → curve is more concentrated (less skewed)
Larger σ → curve becomes more spread and heavily skewed right

In Machine Learning

• Modelling positive, skewed data - Income, prices, durations, transaction amounts
• Feature engineering - Taking log of skewed data often normalises it
• Probabilistic models - Log-normal likelihoods in regression or Bayesian models
• Finance - Stock price modelling under continuous compounding
• Simulation / Monte Carlo - Sampling multiplicative random processes