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Logarithmic Distribution — Definition, Formula & Examples

The logarithmic distribution (also called the log-series distribution) is a discrete probability distribution defined on the positive integers 1, 2, 3, … where the probabilities decrease roughly in proportion to 1/k1/k scaled by powers of a parameter pp. It often arises when modeling the number of species with a given abundance or the number of items in a category.

A discrete random variable XX follows a logarithmic distribution with parameter p∈(0,1)p \in (0,1) if its probability mass function is P(X=k)=−1ln⁡(1−p)⋅pkkP(X = k) = \frac{-1}{\ln(1-p)} \cdot \frac{p^k}{k} for k=1,2,3,…k = 1, 2, 3, \ldots. The name comes from the connection to the Maclaurin series expansion of −ln⁡(1−p)-\ln(1-p), which ensures the probabilities sum to 1.

Key Formula

P(X=k)=−1ln⁡(1−p)⋅pkk,k=1,2,3,…P(X = k) = \frac{-1}{\ln(1-p)} \cdot \frac{p^{k}}{k}, \quad k = 1, 2, 3, \ldots
Where:
  • pp = Parameter of the distribution, where $0 < p < 1$
  • kk = A positive integer outcome (1, 2, 3, …)

How It Works

The single parameter pp controls the shape: values of pp close to 0 concentrate almost all probability on k=1k=1, while values close to 1 spread probability across larger values of kk. The mean is −p(1−p)ln⁡(1−p)\frac{-p}{(1-p)\ln(1-p)} and the variance is −p(p+ln⁡(1−p))(1−p)2(ln⁡(1−p))2\frac{-p(p + \ln(1-p))}{(1-p)^2 (\ln(1-p))^2}. To use the distribution, you estimate pp from data (often via maximum likelihood) and then compute probabilities or expected counts for each integer kk.

Worked Example

Problem: Suppose XX follows a logarithmic distribution with p=0.5p = 0.5. Find P(X=1)P(X = 1) and P(X=2)P(X = 2).
Compute the normalizing constant: First evaluate the constant −1ln⁡(1−p)\frac{-1}{\ln(1-p)}.
−1ln⁡(1−0.5)=−1ln⁡(0.5)=−1−0.6931≈1.4427\frac{-1}{\ln(1-0.5)} = \frac{-1}{\ln(0.5)} = \frac{-1}{-0.6931} \approx 1.4427
Find P(X = 1): Substitute k=1k = 1 into the PMF.
P(X=1)=1.4427⋅0.511=1.4427×0.5≈0.7213P(X=1) = 1.4427 \cdot \frac{0.5^1}{1} = 1.4427 \times 0.5 \approx 0.7213
Find P(X = 2): Substitute k=2k = 2 into the PMF.
P(X=2)=1.4427⋅0.522=1.4427×0.125≈0.1803P(X=2) = 1.4427 \cdot \frac{0.5^2}{2} = 1.4427 \times 0.125 \approx 0.1803
Answer: P(X=1)≈0.7213P(X=1) \approx 0.7213 and P(X=2)≈0.1803P(X=2) \approx 0.1803. Most of the probability mass sits at k=1k=1.

Visualization

Why It Matters

The logarithmic distribution is a building block of the negative binomial distribution — specifically, a Poisson mixture of logarithmic random variables yields a negative binomial. Ecologists use it (via Fisher's log-series model) to describe species abundance data, and it appears in actuarial science when modeling claim frequencies.

Common Mistakes

Mistake: Confusing the logarithmic distribution with the lognormal distribution.
Correction: The logarithmic (log-series) distribution is discrete on positive integers; the lognormal distribution is continuous on positive reals. They are entirely different families.