ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4801 on D5116: Cogeometric random natural number
Subresult of R3205: Probability mass function for geometric random positive integer

Probability mass function for cogeometric random basic natural number

Formulation 0

Let $N \in \text{Cogeometric}(\theta)$ be a D5116: Cogeometric random natural number.
Let $n \in \mathbb{N}$ be a D996: Natural number.

Then \begin{equation} \mathbb{P}(N = n) = \theta (1 - \theta)^n \end{equation}

Proofs

Proof 0

Let $N \in \text{Cogeometric}(\theta)$ be a D5116: Cogeometric random natural number.
Let $n \in \mathbb{N}$ be a D996: Natural number.

Using R3205: Probability mass function for geometric random positive integer, we have for some geometric random number $M$ with parameter $\theta$ \begin{equation} \mathbb{P}(M = n) = \mathbb{P}(N - 1 = n) = \mathbb{P}(N = n + 1) = \theta (1 - \theta)^n \end{equation} $\square$