ThmDex – An index of mathematical definitions, results, and conjectures.
 ▼ Set of symbols ▼ Alphabet ▼ Deduction system ▼ Theory ▼ Zermelo-Fraenkel set theory ▼ Set ▼ Subset ▼ Power set ▼ Hyperpower set sequence ▼ Hyperpower set ▼ Hypersubset ▼ Subset algebra ▼ Subset structure ▼ Measurable space ▼ Measure space ▼ Probability space ▼ Filtered probability space ▼ Random time ▼ Stopping time ▼ Negative binomial random number
Definition D4001
Geometric random positive integer

Let $X_1, X_2, X_3, \dots \in \text{Bernoulli}(\theta)$ each be a D207: Bernoulli random boolean number such that
 (i) $X_1, X_2, X_3, \dots$ is an D2713: Independent random collection (ii) $\theta \in (0, 1]$
A D5748: Random positive integer $G \in \text{Random} \{ 1, 2, 3, \ldots \}$ is a geometric random positive integer with parameter $\theta$ if and only if $$G \overset{d}{=} \min \left\{ N \in \{ 1, 2, 3, \ldots \} : \sum_{n = 1}^N X_n = 1 \right\}$$
Children
 ▶ D5116: Cogeometric random natural number ▶ D4000: Standard exponential random positive real number
Results
 ▶ R4997 ▶ R4996 ▶ R4805: Dual probability distribution function for geometric random positive integer ▶ R4998: Limit of distribution function of geometric random positive integer scaled by reciprocal of index ▶ R4804: Probability distribution function for geometric random positive integer ▶ R3205: Probability mass function for geometric random positive integer