Lognormal Distribution

The lognormal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. If a variable \(X\) follows a lognormal distribution, then \(\ln(X)\) follows a normal distribution with parameters \(\mu\) and \(\sigma\).

In notation: \[ X \sim \text{Lognormal}(\mu, \sigma^2) \quad \text{means} \quad \ln(X) \sim \mathcal{N}(\mu, \sigma^2) \]

Key Characteristics

  • Skewed to the right (positively skewed)
  • Only defined for positive values: \(X > 0\)
  • Commonly used in fields where values must be positive and multiplicative processes are involved (e.g., finance, biology, environmental studies)

Percentiles

To compute percentiles of a lognormal distribution, we first compute the corresponding percentile of the associated normal distribution, then exponentiate:

For the 90th, 95th, and 99th percentiles:

\[ \text{Normal:} \quad \mu + z_p \cdot \sigma \] \[ \text{Lognormal:} \quad e^{\mu + z_p \cdot \sigma} \]

Where \(z_p\) is the percentile value from the standard normal distribution:

  • \(z_{0.90} \approx 1.2816\)
  • \(z_{0.95} \approx 1.6449\)
  • \(z_{0.99} \approx 2.3263\)

Example

Suppose \(\mu = 1\), \(\sigma = 0.5\). To calculate the 95th percentile:

  1. Compute the normal percentile:
    \[ \mu + z_{0.95} \cdot \sigma = 1 + (1.6449)(0.5) = 1.82245 \]
  2. Convert to lognormal:
    \[ e^{1.82245} \approx 6.18 \]

So, the 95th percentile of this lognormal distribution is approximately 6.18.