Probability Distributions > Beta Geometric Distribution

## What is the Beta Geometric Distribution?

The **beta geometric distribution** (also called the *Type I Geometric)* is a compound distribution that is a generalization of the geometric distribution.

While the geometric distribution models the number of failures that will occur before the first observed success in a sequence of independent Bernoulli trials, the beta geometric distribution allows for varying probability of success from trial to trial.

One specific use often cited is with fecundity in the population, i.e., the number of failures before a successful pregnancy. In fact, the model was originally developed by Porter and Park (1964, as cited in [1]) to model this exact scenario: waiting time to conception. The distribution is often used in reliability engineering to model the number of failures that happen before the first observed success and can also be found various population studies and in process control.

The Yule-Simon distribution is a special case of the beta geometric distribution, when β = 1 [2].

## Properties of the beta geometric distribution

The probability of success parameter (*p)*, has a Beta distribution with shape parameters alpha(*α*) and beta(*β*); both shape parameters are positive (*α *> 0 and *β *> 0).

The beta geometric is often used to model the number of failures that will happen in a binomial process before the first success; it can also be used to model the number of times a gambler loses before winning. The probability of success is the mean of the distribution, given by the formula *α* / (*α* + *β*). This particular usage is often called the *shifted beta binomial* and is where the number of failures is shifted by one. This means that the first “failure” is counted as a success.

Interestingly, the beta geometric distribution has both a probability density function (pdf) and probability mass function (PMF).

The PDF is [3]

Where α and β are the shape parameters and *B* is the beta function. When the distribution is shifted to begin at *x* = 0, the PMF is

While the Geometric distribution assumes a constant probability of success for each trial, the Beta-Geometric model extends this by allowing flexibility in how probabilities change from one trial to another.

## References

- American Statistical Association (1988). Proceedings of the Social Statistics Section.
- King, M. (2017). Statistics: A Practical Approach for Process Control Engineers. John Wiley and Sons.
- NIST. BGEPDF. Retrieved July 19, 2023 from: https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/bgepdf.htm