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A fair X takes 0,2. What is Var(X)?

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高三選擇性必修 第三册(A版).pdf · 7.3 · PDF 67 / printed page 62

Revisit first: Conditions, partitions and posterior probability

TOPIC 01

Distribution means, variances and dependence

Compute weighted moments and apply shifts/scales with correct units and dependence conditions.

What you will be able to explain

  • Compute weighted moments and apply shifts/scales with correct units and dependence conditions.
  • Use nonnegative variance to reject impossible summaries and compare equal-mean distributions.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

A fair X takes 0,2. What is Var(X)?

X=(0,2); P=(1/2,1/2)X=(0,2);\ P=(1/2,1/2)
  • Finite-distribution probabilities must be nonnegative and sum to one; dependence is only as explicitly stated.
Skills and prerequisite lessons

Scope: School extension · This task develops the school textbook skill beyond the explicitly listed JAE syllabus topics.

  1. 2
  2. 0
  3. 1
  4. 4
Working and explanation
BUILD THE REASONING
Hint 1
Center before squaring or subtract the squared mean from the second moment.
Hint 2
Start from this intermediate relation.
E(X)=1; E(X2)=2E(X)=1;\ E(X^2)=2
Worked solution
  1. Center before squaring or subtract the squared mean from the second moment.
  2. Establish and simplify this mathematical relation.
    E(X)=1; E(X2)=2E(X)=1;\ E(X^2)=2
  3. Establish and simplify this mathematical relation.
    Var⁡(X)=2−1=1\operatorname{Var}(X)=2-1=1
  4. The variance is not the raw second moment.
1
Checks and common pitfalls: The variance is not the raw second moment.
Think first. Reveal a hint when the class is ready.
Focus on one question
Teacher preparation and assessment
40-minute lesson
MinutesFocusTeaching and student action
5PredictPredict what happens to variance after multiplying all values by −2 and adding 7.
8Build the ideaSample response: shift does not affect spread; the scale squares to four.
10Guided comparisonSample response: X+1−X=1, so the sum has variance zero.
12Independent practice and discussionCompare equal-mean games and audit proposed moments for feasibility.
5Exit and next stepExit: distinguish E(X²) from E(X)² and state the independence requirement for adding variances.
Question sequence
  • Predict what happens to variance after multiplying all values by −2 and adding 7.
  • Sample response: X+1−X=1, so the sum has variance zero.
Board development
  • Average first, center before measuring spread
    E(X)=∑xpx; Var⁡(X)=E(X2)−E(X)2E(X)=\sum xp_x;\ \operatorname{Var}(X)=E(X^2)-E(X)^2
Expected correct and incorrect responses
  • Correct sample: 1
  • Anticipated wrong response: 2 — Uses the second raw moment without centering.
  • Anticipated wrong response: 11 — Scales variance linearly and adds a constant shift.
  • Using unweighted means for unequal probabilities.
  • Adding variances without checking covariance.
Follow-up questions
  • Follow-up: Center before squaring or subtract the squared mean from the second moment. Then explain why 2 fails.
  • Follow-up: Square the scale and remove the constant shift. Then explain why 11 fails.
  • Explain why a negative scale reverses positions but does not make variance negative.
Remediation
  • Sample response: shift does not affect spread; the scale squares to four.
  • Return to the first example, identify the missing condition, then solve one fresh task without viewing its solution.
Exit check
  • Exit: distinguish E(X²) from E(X)² and state the independence requirement for adding variances.
Assessment checklist and sample responses
  • Representation and conditions
    For probabilities 3/4,1/4 on 0,2, E(X)=0·3/4+2·1/4=1/2.
  • Connected mathematical reasoning
    For fair 0,2, the mean is 1 and both squared deviations are 1, so variance is 1.
  • Interpretation and boundary check
    If Y=1−X, X+Y=1 and variance is zero; independence is false.
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Curriculum and source notes ↗