Kenneth ChengMathematical modeling
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Normal distribution

Read the idea, work independently, then explain what changed.

高三選擇性必修 第三册(A版).pdf · 7.5 · PDF 88 / printed page 83

Revisit first: Numerical characteristics of random variables

TOPIC 01

Normal distribution

Standardise a normal variable and interpret supplied table probabilities and symmetry.

What you will be able to explain

  • Standardise a normal variable and interpret supplied table probabilities and symmetry.
  • Justify the method and check the conditions in a new situation.

Defining relation

Standardise a normal variable and interpret supplied table probabilities and symmetry.

Z=(X−μ)/σZ=(X-\mu)/\sigma

Conditions

σ>0; probability table values are supplied in each task. A continuous variable has zero probability at a single point.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Does shifting the mean change the probability within one standard deviation?

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Normal density with μ=0, σ=1; the vertical scale is multiplied by four for display. P(μ−σ≤X≤μ+σ)≈0.6827, unchanged by location or scale.

Normal density with μ=0, σ=1; the vertical scale is multiplied by four for display. P(μ−σ≤X≤μ+σ)≈0.6827, unchanged by location or scale.

Explain: Calculate two valid cases and explain the change using the defining relation.

Transfer: Compare the same standardised interval in two different normal models.

Try it. Leave your reasoning visible.

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

01 / Foundation#Worked example

X~N(t,4). Find its standard deviation.

t=2t=2
  • σ>0; probability table values are supplied in each task. A continuous variable has zero probability at a single point.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
List the possible values and probabilities before computing moments.
Hint 2
Use this intermediate relation.
σ=σ2\sigma=\sqrt{\sigma^2}
Worked solution

  1. List the possible values and probabilities before computing moments.

  2. Apply the stated relation and retain its conditions.

    σ=4=2\sigma=\sqrt4=2

  3. The variance 4 has square root 2.

The requested value is 2.

Checks and common pitfalls: The variance 4 has square root 2.

Think first. Reveal a hint when the class is ready.

02 / Standard#Worked example

X~N(t,9), Φ(2)=0.9772. Find P(X>t+6).

t=3t=3
  • σ>0; probability table values are supplied in each task. A continuous variable has zero probability at a single point.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Express the event with a conditional probability or a disjoint partition.
Hint 2
Use this intermediate relation.
z=6/3=2z=6/3=2
Worked solution

  1. Express the event with a conditional probability or a disjoint partition.

  2. Apply the stated relation and retain its conditions.

    P(X>μ+2σ)=1−Φ(2)=0.0228P(X>\mu+2\sigma)=1-\Phi(2)=0.0228

  3. The requested event is a right tail, not a left cumulative probability.

The requested value is 0.0228.

Checks and common pitfalls: The requested event is a right tail, not a left cumulative probability.

Think first. Reveal a hint when the class is ready.

03 / Transfer#Worked example

Models A~N(t,4), B~N(t+10,9). Compare P(A≤t+2) and P(B≤t+13).

t=4t=4
  • σ>0; probability table values are supplied in each task. A continuous variable has zero probability at a single point.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Express the event with a conditional probability or a disjoint partition.
Hint 2
Standardise each bound.
Worked solution

  1. Express the event with a conditional probability or a disjoint partition.

  2. Apply the stated relation and retain its conditions.

    zA=2/2=1,zB=3/3=1z_A=2/2=1,\quad z_B=3/3=1

  3. Equal z-scores give equal probabilities despite different units and scales.

The requested relation or conclusion is shown below.

P(A≤μA+σA)=P(B≤μB+σB)P(A\le\mu_A+\sigma_A)=P(B\le\mu_B+\sigma_B)

Checks and common pitfalls: Equal z-scores give equal probabilities despite different units and scales.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

X~N(t,4). Find its standard deviation.

t=5t=5
  • σ>0; probability table values are supplied in each task. A continuous variable has zero probability at a single point.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
List the possible values and probabilities before computing moments.
Hint 2
Use this intermediate relation.
σ=σ2\sigma=\sqrt{\sigma^2}
Worked solution

  1. List the possible values and probabilities before computing moments.

  2. Apply the stated relation and retain its conditions.

    σ=4=2\sigma=\sqrt4=2

  3. The variance 4 has square root 2.

The requested value is 2.

Checks and common pitfalls: The variance 4 has square root 2.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Standardise x=t+4 under X~N(t,4). Find z.

t=6t=6
  • σ>0; probability table values are supplied in each task. A continuous variable has zero probability at a single point.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
List the possible values and probabilities before computing moments.
Hint 2
Use this intermediate relation.
z=(x−μ)/σz=(x-\mu)/\sigma
Worked solution

  1. List the possible values and probabilities before computing moments.

  2. Apply the stated relation and retain its conditions.

    z=(10−6)/2=2z=(10-6)/2=2

  3. Divide by standard deviation, not variance.

The requested value is 2.

Checks and common pitfalls: Divide by standard deviation, not variance.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

X~N(t,4), Φ(1)=0.8413. Find P(X≤t+2).

t=7t=7
  • σ>0; probability table values are supplied in each task. A continuous variable has zero probability at a single point.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Express the event with a conditional probability or a disjoint partition.
Hint 2
Use this intermediate relation.
(t+2−t)/2=1(t+2-t)/2=1
Worked solution

  1. Express the event with a conditional probability or a disjoint partition.

  2. Apply the stated relation and retain its conditions.

    P(X≤μ+σ)=Φ(1)=0.8413P(X\le\mu+\sigma)=\Phi(1)=0.8413

  3. The requested bound is one standard deviation above the mean.

The requested value is 0.8413.

Checks and common pitfalls: The requested bound is one standard deviation above the mean.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

X~N(t,4), Φ(1)=0.8413. Find P(t−2≤X≤t+2).

t=8t=8
  • σ>0; probability table values are supplied in each task. A continuous variable has zero probability at a single point.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Express the event with a conditional probability or a disjoint partition.
Hint 2
Use this intermediate relation.
Φ(−1)=1−Φ(1)\Phi(-1)=1-\Phi(1)
Worked solution

  1. Express the event with a conditional probability or a disjoint partition.

  2. Apply the stated relation and retain its conditions.

    P(−1≤Z≤1)=2Φ(1)−1=0.6826P(-1\le Z\le1)=2\Phi(1)-1=0.6826

  3. Subtract the lower tail rather than adding two cumulative probabilities.

The requested value is 0.6826.

Checks and common pitfalls: Subtract the lower tail rather than adding two cumulative probabilities.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

X~N(t,4), Φ(1)=0.8413. Find P(t−2≤X≤t+2).

t=9t=9
  • σ>0; probability table values are supplied in each task. A continuous variable has zero probability at a single point.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Express the event with a conditional probability or a disjoint partition.
Hint 2
Use this intermediate relation.
Φ(−1)=1−Φ(1)\Phi(-1)=1-\Phi(1)
Worked solution

  1. Express the event with a conditional probability or a disjoint partition.

  2. Apply the stated relation and retain its conditions.

    P(−1≤Z≤1)=2Φ(1)−1=0.6826P(-1\le Z\le1)=2\Phi(1)-1=0.6826

  3. Subtract the lower tail rather than adding two cumulative probabilities.

The requested value is 0.6826.

Checks and common pitfalls: Subtract the lower tail rather than adding two cumulative probabilities.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

X~N(t,9), Φ(2)=0.9772. Find P(X>t+6).

t=10t=10
  • σ>0; probability table values are supplied in each task. A continuous variable has zero probability at a single point.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Express the event with a conditional probability or a disjoint partition.
Hint 2
Use this intermediate relation.
z=6/3=2z=6/3=2
Worked solution

  1. Express the event with a conditional probability or a disjoint partition.

  2. Apply the stated relation and retain its conditions.

    P(X>μ+2σ)=1−Φ(2)=0.0228P(X>\mu+2\sigma)=1-\Phi(2)=0.0228

  3. The requested event is a right tail, not a left cumulative probability.

The requested value is 0.0228.

Checks and common pitfalls: The requested event is a right tail, not a left cumulative probability.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

X~N(t,4). Find P(X=t).

t=11t=11
  • σ>0; probability table values are supplied in each task. A continuous variable has zero probability at a single point.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Express the event with a conditional probability or a disjoint partition.
Hint 2
A continuous distribution assigns zero mass to a point.
Worked solution

  1. Express the event with a conditional probability or a disjoint partition.

  2. Apply the stated relation and retain its conditions.

    P(X=μ)=0P(X=\mu)=0

  3. The density is largest at the mean, but point probability is still zero.

The requested value is 0.

Checks and common pitfalls: The density is largest at the mean, but point probability is still zero.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

Models A~N(t,4), B~N(t+10,9). Compare P(A≤t+2) and P(B≤t+13).

t=12t=12
  • σ>0; probability table values are supplied in each task. A continuous variable has zero probability at a single point.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Express the event with a conditional probability or a disjoint partition.
Hint 2
Standardise each bound.
Worked solution

  1. Express the event with a conditional probability or a disjoint partition.

  2. Apply the stated relation and retain its conditions.

    zA=2/2=1,zB=3/3=1z_A=2/2=1,\quad z_B=3/3=1

  3. Equal z-scores give equal probabilities despite different units and scales.

The requested relation or conclusion is shown below.

P(A≤μA+σA)=P(B≤μB+σB)P(A\le\mu_A+\sigma_A)=P(B\le\mu_B+\sigma_B)

Checks and common pitfalls: Equal z-scores give equal probabilities despite different units and scales.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

12 / Transfer#Your turn

X~N(t,4). Find P(X=t).

t=13t=13
  • σ>0; probability table values are supplied in each task. A continuous variable has zero probability at a single point.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Express the event with a conditional probability or a disjoint partition.
Hint 2
A continuous distribution assigns zero mass to a point.
Worked solution

  1. Express the event with a conditional probability or a disjoint partition.

  2. Apply the stated relation and retain its conditions.

    P(X=μ)=0P(X=\mu)=0

  3. The density is largest at the mean, but point probability is still zero.

The requested value is 0.

Checks and common pitfalls: The density is largest at the mean, but point probability is still zero.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

Models A~N(t,4), B~N(t+10,9). Compare P(A≤t+2) and P(B≤t+13).

t=14t=14
  • σ>0; probability table values are supplied in each task. A continuous variable has zero probability at a single point.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Express the event with a conditional probability or a disjoint partition.
Hint 2
Standardise each bound.
Worked solution

  1. Express the event with a conditional probability or a disjoint partition.

  2. Apply the stated relation and retain its conditions.

    zA=2/2=1,zB=3/3=1z_A=2/2=1,\quad z_B=3/3=1

  3. Equal z-scores give equal probabilities despite different units and scales.

The requested relation or conclusion is shown below.

P(A≤μA+σA)=P(B≤μB+σB)P(A\le\mu_A+\sigma_A)=P(B\le\mu_B+\sigma_B)

Checks and common pitfalls: Equal z-scores give equal probabilities despite different units and scales.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • Standardise a normal variable and interpret supplied table probabilities and symmetry.
    • Which condition is essential in normal distribution?
    • Does shifting the mean change the probability within one standard deviation?

    Board plan

    • Defining relation: Standardise a normal variable and interpret supplied table probabilities and symmetry.
      Z=(X−μ)/σZ=(X-\mu)/\sigma
    • Conditions: σ>0; probability table values are supplied in each task. A continuous variable has zero probability at a single point.

    Anticipated thinking

    • The second parameter in N(μ,σ²) is variance, not standard deviation.

    Assessment checklist

    • 1 mark: choose the correct representation and conditions.
    • 1 mark: establish the intermediate relation.
    • 1 mark: complete a connected calculation or proof.
    • 1 mark: interpret and check the conclusion.

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    Curriculum and source notes ↗