The **modified geometric distribution** gives the number of failures until the first success in a series of independent trials. The modified geometric distribution starts at zero—that’s why it’s sometimes called the *zero-modified (or zero-truncated) geometric distribution*; a special case is the zero-modified negative binomial [1].

## Geometric vs. Modified Geometric Distribution

Both the geometric and modified geometric are discrete analogs of the exponential distribution. The difference between a modified geometric distribution and geometric distribution with the same parameter is that the geometric distribution starts at one.

Another way to look at the difference: the geometric distribution models the number of trials until the first “success” in a series of Bernoulli trials, while the modified geometric distribution models the number of trials

*before*the first success: if X – Geom(

*a*), X – 1 – ModGeom(

*a*) [2].

As an example [3], time slices in a computer program finish execution in a given time slice with probability *p*. The number of time slices needed for the program to complete can be modeled with the modified geometric distribution. Another example from computing: The distribution can model the number of visits to a given state within a simple random walk excursion [4].

## References

[1] Mahler, C. (2017). Mahler’s Guide to Frequency Distributions: Exam C. Retrieved November 5, 2021 from: https://www.actexmadriver.com/samples/Mahler_4C-MAH-17SSM-E_sample_11-15-16.pdf

[2] Bagchi, S. Fault-Tolerant Computer System Design. ECE 60872/CS 590. Topic 2: Discrete Distributions.

[3] Ciardo, G. et al. (1994). On the Minimum of Independent Geometrically Distributed Random Variables. NASA from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.844.6616&rep=rep1&type=pdf

[4] Baron, M. & Rukhin, A. (1999). Distribution of the number of visits of a random walk. Retrieved November 5, 2021 from: https://www.tandfonline.com/doi/abs/10.1080/15326349908807552?journalCode=lstm19