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

#### Description

This app models the success-runs Markov chain corresponding to Bernoulli trials with probability of success \(p\). The state at any time is the length of the current run of successes. The limiting distribution is the geometric distribution on \(\N\) with parameter \(1 - p\).

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.