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Expected Value & Martingales

Linearity, indicators, optional stopping, induction.

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Backwards induction: solve from the terminal state
untried
recommend Dynamic programming: memoize the overlapping subproblems first
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Linearity of expectation: decompose, then sum
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First-step analysis: condition on the first transition
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recommend Markov chains: stationary distributions and long-run fractions first
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Indicator random variables: choose the indexing first
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Martingales + Optional Stopping: one line replaces the linear system
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recommend Markov chains: stationary distributions and long-run fractions first
Reflection principle: count bad paths by mirroring them
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recommend Random walk: gambler's ruin and hitting probabilities +1 more first
Variance of sums + law of total variance
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recommend Linearity of expectation: decompose, then sum first
Optimal stopping: the threshold rule
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recommend Backwards induction: solve from the terminal state +1 more first