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Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Use Chebyshev's inequality to obtain a distribution-free variance bound.
- Justify the conclusion "At ε=1, the probability is at most 0.04" using the stated assumptions.
Before you start
Variance of independent 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
If iid observations have variance four, bound P(|X̄−μ|≥1) for n=100.
Why this math matters
Use Chebyshev's inequality to obtain a distribution-free variance bound. 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 have a common finite variance four.
- Independence justifies the displayed variance reduction.
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.
Prove concentration of an average without a normal model
PausedQuestion: Start with the question. Paused.
Question
Start with the question
If iid observations have variance four, bound P(|X̄−μ|≥1) for n=100.
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
Var(X̄)=4/100=0.04
Independence reduces the sample-mean variance.
Work through the mathematics
P(|X̄−μ|≥ε)≤Var(X̄)/ε²
Chebyshev uses only a finite second moment.
Check the conclusion
At ε=1, the probability is at most 0.04
Letting n grow makes the same bound tend to zero, illustrating the weak law of large numbers.
The result
At ε=1, the probability is at most 0.04
Letting n grow makes the same bound tend to zero, illustrating the weak law of large numbers.
Common mistakes to catch
- A conservative bound is not an exact tail probability.
- Finite mean alone is insufficient for this variance-based calculation.
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
How large must n be for this bound to be at most 0.01?
Show a hint
Solve 4/n≤0.01.
Reveal answer and explanation
n≥400
The inequality is a sufficient sample-size bound.
Practice 2
Does a 0.04 bound mean the probability equals 0.04?
Show a hint
An inequality need not be sharp for a given law.
Reveal answer and explanation
No
The actual probability may be much smaller.
Take the idea with you
Choose a defensible sample size when the population shape is unknown but variance is bounded.
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: Estimate a rate by maximizing its likelihood
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