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Modified Geometric Distribution

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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.

modified geometric distribution

The modified geometric distribution can model a random walk confined to a lattice.

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.
[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

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Stephanie Glen. "Modified Geometric Distribution" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/modified-geometric-distribution/
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