Geometric Distribution

The geometric distribution, also known as the discrete waiting time distribution, models the number of failures before the first success in a series of Bernoulli trials. A Bernoulli trial is an experiment with two possible outcomes, labeled 'success' and 'failure,' where the probability of success has a fixed value for all trials.

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

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

The cumulative distribution function (CDF) is:

$$F(k)=1-(1-p{)}^{k+1}$$

The domain of the geometric distribution is $k\in \{0,1,2,\dots \}$. The parameter $p$ must satisfy $0\le p\le 1$.

The geometric distribution is widely used in various fields due to its simplicity and fundamental nature. Common applications include:

• The number of successive hits by a baseball player (assuming the probability of a hit is constant) has a geometric distribution with parameter $p=1-\text{(batting average)}$.

• The number of attempts made by a player to hit a target has a geometric distribution.

The geometric distribution has several important statistical properties:

The geometric distribution is closely related to several other distributions:

• The Negative Binomial Distribution is a generalization of the geometric distribution, modeling the number of failures before the $n$th success in a series of Bernoulli trials.

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

The geometric distribution is implemented by the GeometricDistribution class. It has one constructor which takes one argument: the probability of success of a trial. The probability must be between 0 and 1. The following constructs a geometric distribution with a probability of success 0.4:

var geometric = new GeometricDistribution(0.4);

Dim geometric = New GeometricDistribution(0.4)

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

let geometric = GeometricDistribution(0.4)

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

GeometricDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator.

var random = new Pcg32();
int sample = GeometricDistribution.Sample(random, 0.4);

Dim random = New Pcg32()
Dim sample = GeometricDistribution.Sample(random, 0.4)

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

let random = Pcg32()
let sample = GeometricDistribution.Sample(random, 0.4)

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