The log-normal distribution is a continuous probability distribution whose logarithm is normally distributed. It models positive-valued quantities that arise as the product of many independent positive factors.
A random variable \(X\) follows a log-normal distribution with parameters \(\mu \in \mathbb{R}\) and \(\sigma > 0\) if
\[ X \sim \text{LogNormal}(\mu, \sigma^2) \quad \text{and} \quad \ln(X) \sim \mathcal{N}(\mu, \sigma^2). \]
The probability density function is
\[ f(x) = \frac{1}{x\sigma\sqrt{2\pi}} \exp\!\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right), \quad x > 0. \]
Interpretation: The parameters \(\mu\) and \(\sigma\) are the mean and standard deviation of \(\ln(X)\), not of \(X\) itself. The density is right-skewed with a long upper tail.
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from scipy.stats import lognorm
sns.set_style('whitegrid')
sns.set_palette('Set2')
x = np.linspace(0, 5, 400)
fig, ax = plt.subplots(figsize=(7, 4))
for sigma in (0.25, 0.5, 1.0):
ax.plot(x, lognorm.pdf(x, sigma),
lw=2, label=rf'$\sigma = {sigma}$')
ax.set_xlabel('$x$')
ax.set_ylabel('$f(x)$')
ax.set_title(r'Log-normal density ($\mu = 0$)')
ax.legend()
plt.show()
The cumulative distribution function is
\[ F(x) = \Phi\!\left(\frac{\ln x - \mu}{\sigma}\right), \quad x > 0, \]
where \(\Phi\) is the CDF of the standard normal distribution.
The expectation and variance of \(X \sim \text{LogNormal}(\mu, \sigma^2)\) are
\[ \mathbb{E}[X] = \exp\!\left(\mu + \frac{\sigma^2}{2}\right) \quad \& \quad \operatorname{Var}(X) = \left(e^{\sigma^2} - 1\right) e^{2\mu + \sigma^2}. \]
More generally, every moment exists and equals \(\mathbb{E}[X^n] = \exp(n\mu + n^2\sigma^2/2)\).
Proof. Writing \(X = e^Y\) with \(Y \sim \mathcal{N}(\mu, \sigma^2)\) and using the normal MGF \(M_Y(t) = e^{\mu t + \sigma^2 t^2/2}\),
\[ \mathbb{E}[X^n] = \mathbb{E}[e^{nY}] = M_Y(n) = \exp\!\left(n\mu + \frac{n^2\sigma^2}{2}\right). \]
The stated mean and variance are the cases \(n = 1, 2\).
Note: Although all moments are finite, the moment generating function of \(X\) does not exist for any \(t > 0\), because the tail is too heavy.
If \(Y \sim \mathcal{N}(\mu, \sigma^2)\) then \(X = e^Y \sim \text{LogNormal}(\mu, \sigma^2)\), and conversely \(\ln(X) \sim \mathcal{N}(\mu, \sigma^2)\). This exponential link, together with a multiplicative Central Limit Theorem, makes the log-normal the natural limit of products of independent positive factors.
The distribution is right-skewed, and its median lies below its mean:
\[ \operatorname{Median}(X) = e^\mu < e^{\mu + \sigma^2/2} = \mathbb{E}[X]. \]
Stock prices and asset returns are often modelled as log-normal, as in geometric Brownian motion.
Pollutant concentrations, particle sizes, and precipitation amounts frequently follow log-normal distributions.
Times to failure under multiplicative degradation, and incubation periods of diseases, are commonly log-normal.