\(\newcommand{\N}{\mathbb{N}}\)

### Remaining-Life Experiment

#### Description

This app models the remaining-life Markov chain corresponding to the geometric distribution with parameter \(p\). When the chain is in state 0, the next state is \( x \in \N \) with probability \( (1 - p) p^x \). When the chain is in state \( x \in \N_+ \), the next state is \( x - 1 \) deterministically. The given geometric distribution is also the limiting distribution of the chain.

The states are shown in the top graph, with the current state colored red. As the process runs, the chain moves from state to state. The time and the sate are recorded at each update in the record table. The proportion of time that the chain is in each state is shown visually in the graph box in red and displayed numerically in the distribution table. The limiting distribution is shown visually in the graph box in blue and displayed numerically in the distribution table. The parameter \(p\) and the initial state \(x_0\) can be varied with the input controls.