A Generalized Geometric Distribution from Vertically Dependent Bernoulli Random Variables

A Generalized Geometric Distribution from Vertically Dependent Bernoulli Random Variables

 For full proofs and derivations, read here.

Conclusion

The standard geometric distribution counts the number of independent Bernoulli trials until the first success. This paper uses the works of Korzeniowski, and Traylor and Hathcock [2, 4, 5] on dependent Bernoulli and categorical random variables to develop a generalized geometric distribution built from dependent Bernoulli random variables. The main result of the paper is that the pdf for the generalized geometric distribution is independent of the dependency structure of the Bernoulli random variables that comprise it. That is, regardless of dependency structure, the pdf for the generalized geometric distribution is given by Proposition 1. Various properties and characterizations were given, including the moment generating function, mean, variance, skew, and entropy. The effect of dependency on each property was studied.
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References

  1. Skewness. https://en.wikipedia.org/wiki/skewness.
  2. Andrzej Korzeniowski. On correlated random graphs. Journal of Probability and Statistical Science,pages 43–58, 2013.
  3. Claude Shannon. A mathematical theory of communication. The Bell System Technical Journal, 1948.
  4.  Rachel Traylor. A generalized multinomial distribution from dependent categorical random variables. Academic Advances of the CTO, 2017.
  5. Rachel Traylor and Jason Hathcock. Vertical dependency in sequences of categorical random variables. Academic Advances of the CTO, 2017.