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Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Compute conditional expectation as a measurable groupwise average.
- Justify the conclusion "E[X|G]=(2,2,6,6)" using the stated assumptions.
Before you start
Sigma-algebras and expectations.
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
On four equally likely outcomes let X=(1,3,2,10), and reveal only the partition {1,2},{3,4}. Find E[X|G].
Why this math matters
Compute conditional expectation as a measurable groupwise average. 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:
- All four outcomes have probability one quarter.
- X is square-integrable in this finite model.
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.
Project a random variable onto partial information
PausedQuestion: Start with the question. Paused.
Question
Start with the question
On four equally likely outcomes let X=(1,3,2,10), and reveal only the partition {1,2},{3,4}. Find E[X|G].
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
Average X on the first atom: (1+3)/2=2
A G-measurable estimate must be constant on that atom.
Work through the mathematics
Average X on the second atom: (2+10)/2=6
The other observable atom receives its own conditional mean.
Check the conclusion
E[X|G]=(2,2,6,6)
Integrals agree with X on each G-event; also E[E[X|G]]=4=E[X].
The result
E[X|G]=(2,2,6,6)
Integrals agree with X on each G-event; also E[E[X|G]]=4=E[X].
Common mistakes to catch
- Conditional expectation is a random variable, not always one number.
- Conditioning on finer information changes the allowable measurable functions.
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
What is E[X|trivial information]?
Show a hint
Only constant measurable estimates are allowed.
Reveal answer and explanation
Four everywhere
It is the unconditional mean.
Practice 2
Why is X−E[X|G] orthogonal to G-measurable square-integrable variables?
Show a hint
Its conditional mean is zero.
Reveal answer and explanation
The expected product vanishes
Conditioning the product extracts the measurable factor and leaves conditional mean zero.
Take the idea with you
Treat a coarse sensor's best squared-error estimate as a projection onto its information.
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: Check fairness through conditional expectation
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