Beta Geometric Distribution

< List of probability distributions

What is the Beta Geometric Distribution?

The beta geometric distribution (also called the Type I Geometric) is a type of geometric distribution, where 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). It is a type of compound distribution.

Examples of the beta geometric distribution PMF. Image:

The distribution often models the number of failures that will happen in a binomial process before the first observed success. 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.

One particular 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 American Statistical Association., 1988) to model this exact scenario: waiting time to conception. It is also used in various other population studies and in process control.

Geometric vs. Beta Geometric Distribution

The main difference between the geometric and the Beta Geometric is that p remains constant with the geometric and changes with the beta geometric.


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 November 12, 2019 from:
R Documentation. Betageom. Retrieved November 12, 2019 from: