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Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Verify a stopped expectation at a bounded random time.
- Justify the conclusion "E[Sτ]=1/2+0−2/4=0" using the stated assumptions.
Before you start
Martingales and stopping times.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
For a fair ±1 walk, stop at τ=1 if the first step is +1, otherwise stop at τ=2. Compute E[Sτ].
Why this math matters
Verify a stopped expectation at a bounded random time. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The increments are independent fair signs.
- The stopping decision uses only information already observed.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Use optional stopping only with its conditions
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For a fair ±1 walk, stop at τ=1 if the first step is +1, otherwise stop at τ=2. Compute E[Sτ].
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
P(Sτ=1)=1/2
A positive first step ends the walk immediately.
Work through the mathematics
P(Sτ=0)=1/4 and P(Sτ=−2)=1/4
A negative first step is followed by either possible second step.
Check the conclusion
E[Sτ]=1/2+0−2/4=0
The direct calculation agrees with optional stopping because τ≤2 is bounded.
The result
E[Sτ]=1/2+0−2/4=0
The direct calculation agrees with optional stopping because τ≤2 is bounded.
Common mistakes to catch
- A random time that depends on future information may not be a stopping time.
- Almost-sure finiteness is not identical to boundedness.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Is the first time a fair walk reaches +1 bounded?
Show a hint
It can take arbitrarily many steps.
Reveal answer and explanation
No
A finite deterministic upper bound does not exist.
Practice 2
Can the bounded stopping theorem be applied to that hitting time without more work?
Show a hint
Check the theorem's hypothesis.
Reveal answer and explanation
No
Unbounded stopping times require additional conditions and can defeat naive expectation arguments.
Take the idea with you
Explain why a stopping rule alone cannot justify a claimed guaranteed positive expected gain.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: See the extra term in stochastic differentiation
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