Why Lognormal Distribution Is Key to Modeling Rainfall and Financial Data

Rainy season often shows uneven daily precipitation, with some days experiencing heavy downpours while others have only light drizzle.

Modeling this variability requires a suitable statistical model, and the lognormal distribution is commonly employed.

This article introduces the mathematical expression, key properties, and applications of the lognormal distribution, followed by a concrete case study on rainfall modeling.

Definition and Properties

Lognormal distribution is a random variable whose logarithm follows a normal distribution.

If the logarithm of a random variable follows a normal distribution with mean μ and variance σ², then the variable follows a lognormal distribution with parameters derived from μ and σ.

Its probability density function is:

(formula omitted)

Important properties:

Non‑negativity : the variable can only take positive values.

Skewness : it is right‑skewed, with most data concentrated on the left and a long tail on the right.

Log‑normality : taking the logarithm of the variable yields a normal distribution.

Scale invariance : multiplying the variable by a constant preserves the lognormal form.

Applications

The lognormal distribution is used in many fields, especially finance, environmental science, and biology.

In finance, asset‑price models such as the Black‑Scholes option pricing assume that stock prices follow a lognormal distribution.

Many natural phenomena like rainfall amount and river flow are also modeled as lognormal; in biology, cell size and animal weight often follow a lognormal distribution.

Case Study: Rainfall Modeling

This section analyzes a concrete example of applying the lognormal distribution to daily rainfall amounts in a region.

Assume the logarithm of daily rainfall follows a normal distribution with known mean and variance; then the daily rainfall follows a lognormal distribution.

The probability density function of daily rainfall is:

(formula omitted)

Based on this distribution, we simulate future precipitation:

Its empirical frequency distribution compared with the theoretical curve:

The lognormal distribution captures the data’s non‑negativity and skewness, reflecting actual rainfall characteristics. Understanding rainfall distribution aids water‑resource management and disaster mitigation, allowing more accurate prediction of extreme events and better response strategies.