Log-Series Distribution

The log-series distribution, also known as the logarithmic series distribution or the logarithmic distribution, is a discrete probability distribution derived from the MacLaurin series of a logarithmic function. It is used to model the number of occurrences of rare events.

The log-series distribution has one parameter $p$ (0 < $p$ < 1), which represents the probability of success. The probability mass function (PMF) is given by:

$$P(X=k)=-\frac{p^k}{k\log (1-p)}$$

The cumulative distribution function (CDF) is:

$$F(k)=1+\frac{p}{\log (1-p)}\sum _{i=1}^k\frac{p^i}{i}$$

The domain of the log-series distribution is $k\in \{1,2,3,\dots \}$. The parameter $p$ must satisfy $0<p<1$.

The log-series distribution is widely used in various fields due to its ability to model the number of occurrences of rare events. Common applications include:

• Modeling the number of species in ecological studies where most species are rare.

• Analyzing word frequencies in natural language processing, where most words are infrequent.

The log-series distribution has several important statistical properties:

The log-series distribution is closely related to several other distributions:

• The Bernoulli Distribution is a special case of the binomial distribution with $n=1$.

• The Binomial Distribution models the number of successes in a fixed number of Bernoulli trials with replacement.

The log-series distribution is implemented by the LogarithmicSeriesDistribution class. It has one constructor that takes one argument: the probability of success. The following constructs a log-series distribution for $p=0.5$:

var logSeries = new LogarithmicSeriesDistribution(0.5);

Dim logSeries = New LogarithmicSeriesDistribution(0.5)

No code example is currently available or this language may not be supported.

let logSeries = new LogarithmicSeriesDistribution(0.5);

The LogarithmicSeriesDistribution class has one specific property, ProbabilityOfSuccess, which returns the probability of success of a trial.

LogarithmicSeriesDistribution has one static (Shared in Visual Basic) method, Sample(Random), which generates a random sample using a user-supplied uniform random number generator. It has two overloads, corresponding to each of the two constructors.

var random = new Pcg32();
int sample = LogarithmicSeriesDistribution.Sample(random, 0.5);

Dim random = New Pcg32()
Dim sample = LogarithmicSeriesDistribution.Sample(random, 0.5)

No code example is currently available or this language may not be supported.

let random = new Pcg32()
let sample = LogarithmicSeriesDistribution.Sample(random, 0.5)

The above example uses the Pcg32 class to generate uniform random numbers.

For details of the properties and methods common to all discrete probability distribution classes, see the topic on Discrete Probability Distributions.

• "Introduction to Probability Models" by Sheldon M. Ross.

• "Probability and Statistics" by Morris H. DeGroot and Mark J. Schervish.

Other Resources