# Modified geometric distribution

The modified geometric distribution provides the number of failures until the first success in a set of independent Bernoulli trials. The distribution, which starts at zero, is also known as the zero-modified (or zero-truncated) geometric distribution. Note that a zero-modified negative binomial is a specific example of this distribution [1].

The modified geometric distribution (MGD) and the “regular” geometric distribution are discrete analogs of the exponential distribution. The difference between these two distributions with the same parameter is that while the modified geometric starts at zero, the geometric distribution starts at one:

if X – Geom(a), X – 1 – ModGeom(a) [2].

This gives better flexibility in modeling where a “success” happens: if you want to model the number of trials until the first “success”, use the regular geometric; if you want to model the number of trials before the first success, use the modified geometric distribution.

A random variable X follows a modified geometric geometric distribution if its probability mass function (PMF) is [3]

## History of the modified geometric distribution

The modified geometric distribution was first introduced by Russian mathematician Andrey Markov in 1892. Markov developed the distribution to model the probability of observing a certain number of failures before the first success in a sequence of Bernoulli trials, which are a series of binary experiments with two possible outcomes.

The modified geometric distribution was later rediscovered by American statistician Frank Ramsey in the early 1930s and independently by mathematician Harold Hotelling around the same time. Ramsey studied the distribution’s properties and its relationship to other distributions, while Hotelling applied it to his work in anthropometry, the study of human physical characteristics.

The modified geometric distribution has found applications in various fields, such as reliability theory, queuing theory, and genetics. It is a useful tool for modeling events that require multiple trials with a binary outcome, and researchers continue to use and study the distribution in modern statistical analysis. For example, the MGD can model time to completion for a computer program when the number of time slices that finish execution in a given time slice have probability p [4]. The distribution can also model the number of visits to a given state within a simple random walk excursion [5].

## 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. Retrieved Jan 1, 2023 from: http://fault-tolerant%20computer%20system%20design%20ece%2060872/CS%20590

[3] Iyer, R. Important discrete distributions: Poisson, geometric and modified geometric. Retrieved May 13, 2023 from: https://courses.engr.illinois.edu/ece313/sp2017/sectionG/Lectures/lec_10.pdf

[4] 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

[5] 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

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