1. **Stating the problem:** Define what a Markov Process is and provide an example. 2. **Definition:** A Markov Process is a type of stochastic process that satisfies the Markov property, meaning the future state depends only on the current state and not on the sequence of events that preceded it. 3. **Markov Property formula:** $$P(X_{n+1} = x | X_n = x_n, X_{n-1} = x_{n-1}, \ldots, X_0 = x_0) = P(X_{n+1} = x | X_n = x_n)$$ This means the conditional probability of the next state depends only on the present state. 4. **Explanation:** This property simplifies the analysis of stochastic processes because the entire history is summarized by the current state. 5. **Example:** Consider a weather model with two states: Sunny (S) and Rainy (R). The probability of tomorrow's weather depends only on today's weather. Transition probabilities: - $P(S \to S) = 0.8$ - $P(S \to R) = 0.2$ - $P(R \to S) = 0.4$ - $P(R \to R) = 0.6$ If today is Sunny, the probability that tomorrow is Sunny is 0.8, regardless of the weather before today. 6. **Summary:** A Markov Process models systems where the next state depends only on the current state, not the full history, making it useful in many fields like physics, finance, and biology.