A Markov Chain is a mathematical model that describes a sequence of events where the probability of each event depends only on the state of the previous event. This fundamental property is known as the memoryless property, meaning the system has no memory of how it arrived at its current state.
In a Markov Chain, transitions between states occur with specific probabilities. For example, in a weather model, if today is sunny, there might be a 70% chance it stays sunny tomorrow, a 20% chance it becomes rainy, and a 10% chance it becomes cloudy. These probabilities are organized in a transition matrix where each row sums to one.
A classic example of a Markov Chain is a random walk. Imagine a particle on a number line that can move left or right with equal probability at each time step. The particle's next position depends only on where it is now, not on how it got there. This demonstrates the memoryless property perfectly.
An important property of Markov Chains is the stationary distribution. After running for a long time, the probabilities of being in each state stabilize to fixed values. This steady-state distribution satisfies the equation pi equals pi times P, where pi is the stationary distribution and P is the transition matrix.
Markov Chains have countless applications across many fields. In finance, they model stock price movements and market behavior. In biology, they describe population dynamics and genetic sequences. Computer scientists use them in algorithms and machine learning, particularly in Hidden Markov Models. They're also essential in physics for molecular dynamics and in economics for market analysis. The memoryless property makes Markov Chains incredibly versatile for modeling systems where the future depends only on the present state.