- Why does the simulated percentage of sunny days eventually settle to a fixed value?
- It settles to the stationary distribution, π. This is a long-run equilibrium where the proportion of time the chain spends in each state becomes constant. The transition probabilities dictate this value; changing the probability of rain after a sunny day, for instance, will change the long-run sunny percentage. The simulator shows this convergence empirically and calculates π theoretically from the matrix equation πP = π.
- Is the 'memoryless' property realistic for real weather?
- It is a simplification. Real weather has memory beyond one day; a rainy pattern might persist for a week due to a large storm system. The one-step Markov model is a useful first approximation for some phenomena and a crucial pedagogical tool for understanding more complex models, like higher-order Markov chains or hidden Markov models, which are used in sophisticated weather forecasting and many other applications.
- What do the arrows and numbers on the state diagram represent?
- The circles are the states (Sunny, Rainy). The arrows show possible transitions between states. The number on each arrow is the conditional transition probability—the probability of moving to the state the arrow points to, given you are currently in the state the arrow comes from. All probabilities leaving a state must sum to 1. These numbers are the entries of the transition probability matrix P.
- Can the chain get 'stuck' in one state forever?
- In this basic two-state model with all probabilities between 0 and 1 (non-extreme), it cannot get permanently stuck. You will always have a chance to transition to the other state. However, if a transition probability is set to 0 (e.g., P(S→R)=0), then the 'Sunny' state becomes absorbing—once sunny, always sunny—and the chain's long-term behavior changes fundamentally. This simulator typically models ergodic chains without absorbing states.