Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Graduate chapters and video availability01 · Read and understand
What you will learn
- Apply factorization to identify a sufficient statistic.
- Justify the conclusion "T is sufficient for p in this model" using the stated assumptions.
Before you start
Likelihoods and iid Bernoulli sampling.
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 iid Bernoulli(p) data x₁,…,xₙ, show why T=Σxᵢ is sufficient for p.
Why this math matters
Apply factorization to identify a sufficient statistic. 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 observations are iid with a common p.
- The Bernoulli model is the specified statistical family.
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.
Keep the data summary a likelihood actually uses
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For iid Bernoulli(p) data x₁,…,xₙ, show why T=Σxᵢ is sufficient for p.
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
L(p;x)=p^(Σxᵢ)(1−p)^(n−Σxᵢ)
Multiplication collects the data dependence through the success count.
Work through the mathematics
L(p;x)=gₚ(T(x))h(x), with h(x)=1 on {0,1}ⁿ
The factorization separates parameter dependence from any remaining data detail.
Check the conclusion
T is sufficient for p in this model
Conditional on T, sequences with the same number of successes have a distribution independent of p.
The result
T is sufficient for p in this model
Conditional on T, sequences with the same number of successes have a distribution independent of p.
Common mistakes to catch
- Sufficiency is model-relative.
- A sufficient statistic need not be a lossless encoding of the sample.
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
For n=4 and T=2, how many sequences are possible?
Show a hint
Choose the success positions.
Reveal answer and explanation
Six
C(4,2)=6, each conditionally equally likely.
Practice 2
Does sufficiency mean the count determines the original sequence?
Show a hint
Many arrangements share a count.
Reveal answer and explanation
No
It preserves parameter-relevant information under the model, not every data detail.
Take the idea with you
Explain why a summary can be adequate for one parameter model but inadequate after that model changes.
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: Update a probability distribution over an unknown probability
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.