Lognormal Distribution¶

Formula¶

\[ X \text{ is lognormal } \iff \log X \sim \mathcal{N}(\mu,\sigma^2) \]

Plot¶

fn: exp(-(log(x)^2)/2)/(x*sqrt(2*PI))
xmin: 0.05
xmax: 4
ymin: 0
ymax: 1.0
height: 280
title: Lognormal PDF (mu=0, sigma=1)

Parameters¶

\(X>0\): positive random variable
\(\mu,\sigma^2\): mean/variance of \(\log X\)

What it means¶

A variable is lognormal if its logarithm is normally distributed, yielding a positively skewed distribution.

What it's used for¶

Modeling positive quantities with multiplicative effects (sizes, incomes, latencies).
Noise models in some scientific/engineering settings.

Key properties¶

Support is strictly positive.
Typically right-skewed.

Common gotchas¶

\(\mu,\sigma\) refer to log-space, not the mean/std of \(X\).
Arithmetic means can be dominated by the tail.

Example¶

If \(\log X\sim \mathcal{N}(0,1)\), then \(X\) is lognormal and always positive.

How to Compute (Pseudocode)¶

Input: distribution parameters and query values (for PMF/PDF/CDF or moments)
Output: distribution quantities

validate parameters
for each query value x (or count k):
  evaluate the PMF/PDF/CDF formula from the card
optionally compute moments/statistics from known closed forms or by summation/integration
return the requested values

Complexity¶

Time: Typically \(O(q)\) for \(q\) query values once parameters are known (assuming constant-time formula evaluation per query)
Space: \(O(q)\) for output values (or \(O(1)\) for a single query)
Assumptions: Parameter estimation/fitting cost is excluded; numerical special-function evaluation can affect constants for some families