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Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Compute a Radon–Nikodym derivative on a finite sample space.
- Justify the conclusion "EQ[X]=4=EP[wX]" using the stated assumptions.
Before you start
Probability measures and weighted sums.
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 {a,b}, let P=(1/4,3/4), Q=(1/2,1/2). Find dQ/dP and use it to average X(a)=2,X(b)=6.
Why this math matters
Compute a Radon–Nikodym derivative on a finite sample space. 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 sample space is finite.
- Q is absolutely continuous with respect to P in the main example.
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.
Change probability measures with likelihood weights
PausedQuestion: Start with the question. Paused.
Question
Start with the question
On {a,b}, let P=(1/4,3/4), Q=(1/2,1/2). Find dQ/dP and use it to average X(a)=2,X(b)=6.
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
w(a)=Q(a)/P(a)=2; w(b)=2/3
Positive P mass at both outcomes permits these density ratios.
Work through the mathematics
EP[wX]=(1/4)·2·2+(3/4)·(2/3)·6
Reweighting changes which measure supplies the average.
Check the conclusion
EQ[X]=4=EP[wX]
The weights also satisfy EP[w]=1, confirming normalization.
The result
EQ[X]=4=EP[wX]
The weights also satisfy EP[w]=1, confirming normalization.
Common mistakes to catch
- A density ratio need not be bounded by one.
- Direction matters: dQ/dP is not dP/dQ.
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 fails if P(b)=0 but Q(b)>0?
Show a hint
No finite density can assign new mass to a P-null event.
Reveal answer and explanation
Q is not absolutely continuous with respect to P
A Radon–Nikodym density with this reference measure cannot exist.
Practice 2
What is EP[w]?
Show a hint
Sum the weighted probabilities.
Reveal answer and explanation
One
The density converts total P mass into total Q mass.
Take the idea with you
Interpret importance weights as correcting for a changed sampling distribution.
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: Keep the data summary a likelihood actually uses
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