ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4996 on D4001: Geometric random positive integer
Subresult of R4998: Limit of distribution function of geometric random positive integer scaled by reciprocal of index
Formulation 0
Let $G_n \in \text{Geometric}(\theta_n)$ be a D4001: Geometric random positive integer for each $n \in \mathbb{N}$ such that
(i) \begin{equation} \lim_{n \to \infty} \theta_n n = 1 \end{equation}
Let $a \in (0, \infty)$ be a D5407: Positive real number.
Then \begin{equation} \lim_{n \to \infty} \mathbb{P} \left( \frac{G_n}{n} \leq a \right) = 1 - e^{-a} \end{equation}
Subresults
▶ R4997
Proofs
Proof 0
Let $G_n \in \text{Geometric}(\theta_n)$ be a D4001: Geometric random positive integer for each $n \in \mathbb{N}$ such that
(i) \begin{equation} \lim_{n \to \infty} \theta_n n = 1 \end{equation}
Let $a \in (0, \infty)$ be a D5407: Positive real number.