geometric distribution

geometric distribution (plural geometric distributions)

1. Either of two slightly different discrete probability distributions, each based on repetitions of a trial with "success" probability p: (1) the number X of trials required to obtain one success, or (2) the number Y = X − 1 of failed trials before the first success.
   • 1973, Elizabeth M. Gammon, “A Syntactical Analysis of Some First-Grade Readers”, in K. J. J. Hintikka, P. Suppes, J.M.E. Moravcsik, editors, Approaches to Natural Language: Proceedings of the 1970 Stanford Workshop on Grammar and Semantics, Springer, page 122:

     If S₁ had been chosen to be a geometric distribution, no a priori choices would have been necessary. This distribution was effective in Suppes (1970), but it would provide a poor fit in this case. If the geometric distribution had been used here, the parameter S₁ would have appeared in every theoretical probability involving N, and whenever one or more adjectives were included, a corresponding number of S₂ = 1 - S₁ terms would have appeared in the probability.

   • 2009, Matthew J. Hassett, Donald Stewart, Probability for Risk Management, 2nd Edition, ACTEX Publications, page 139,

     Recall that we have already calculated the mean and variance for the geometric distribution case (r = 1) in Example 5.17.

   • 2010, Peter A. Rogerson, Statistical Methods for Geography: A Student's Guide, 3rd edition, SAGE Publications, page 92:

     The exponential distribution has a number of other interesting characteristics and properties. It is very similar to the geometric distribution, and in fact is the continuous version of that discrete distribution. Recall that the geometric distribution was used to model the time until the first success, where there were a discrete number of trials.

• The two versions of the distribution differ in their range. Version (1) counts all trials, including the successful one, and has values ≥ 1. Version (2) counts only failures, and has values ≥ 0. The first is sometimes called the shifted geometric distribution, but it is considered better style to explicitly mention the difference in range.
• It is up to the writer to decide which is the geometric distribution.
• The distribution is a special case of the negative binomial distribution, which counts trials needed to reach n successes. It too is used in different versions.
• Sometimes, a writer may define the count as being made until the first "failure". This is logically equivalent, however, and the difference lies only in the interpretation of "success".