To get the most out of the book and its solutions, focus heavily on these foundational sections: Chapter 4: Conditional Expectation
Search for "David Williams Probability with Martingales [Exercise Number]".
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Can I sketch a simple tree diagram to see how the conditional expectations behave?
Once you look up a solution on GitHub or StackExchange, close the tab. Try to rewrite the entire proof from scratch in your own notebook. This ensures you actually understand the logic. If you want to focus on a specific topic, tell me: Which chapter or exercise number are you stuck on? What specific concept (e.g., stopping times, -algebras) needs clarification? To get the most out of the book
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: Check with your university library’s digital repository or ask a course instructor. Some professors keep a copy for teaching assistants. Once you look up a solution on GitHub
While there is no single "official" student solution manual published by the author, the best resources for solutions to Probability with Martingales
By the martingale property, we have $\mathbbE[X_n+1 | \mathcalF_n] = X_n$. Taking expectations, we get: