Kenneth ChengMathematical modeling
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Numerical characteristics of random variables

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

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

Revisit first: Discrete random variables and distributions

TOPIC 01

Numerical characteristics of random variables

Compute expectation and variance and distinguish transformation from random independence.

What you will be able to explain

  • Compute expectation and variance and distinguish transformation from random independence.
  • Justify the method and check the conditions in a new situation.

Defining relation

Compute expectation and variance and distinguish transformation from random independence.

E(X)=∑xpx;Var⁡(X)=E(X2)−[E(X)]2E(X)=\sum xp_x;\quad\operatorname{Var}(X)=E(X^2)-[E(X)]^2

Conditions

Variance is nonnegative; variance of a sum includes covariance unless independence is justified.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Does doubling all outcomes double or quadruple the variance?

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

P(X=0,1,2)=(1/4,1/2,1/4); Y=aX+b has E(Y)=2, Var(Y)=2. Plotted heights represent six times the probability; the horizontal display scale adjusts to include every outcome. At a=0, one outcome b has probability 1.

P(X=0,1,2)=(1/4,1/2,1/4); Y=aX+b has E(Y)=2, Var(Y)=2. Plotted heights represent six times the probability; the horizontal display scale adjusts to include every outcome. At a=0, one outcome b has probability 1.

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

Transfer: Compare X, 2X and X+constant from the same distribution.

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

Find E(X).

X=0,2;P=1/2,1/2X=0,2;\quad P=1/2,1/2
  • Variance is nonnegative; variance of a sum includes covariance unless independence is justified.
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.
E(X)=0/2+t/2E(X)=0/2+t/2
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    E(X)=2/2E(X)=2/2

  3. Expectation is a weighted average, not necessarily a possible outcome.

The requested value is 1.

Checks and common pitfalls: Expectation is a weighted average, not necessarily a possible outcome.

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

02 / Standard#Worked example

Find Var(3X+2).

Var(X)=3Var(X)=3
  • Variance is nonnegative; variance of a sum includes covariance unless independence is justified.
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.
Var(aX+b)=a2Var(X)Var(aX+b)=a^2Var(X)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    Var(3X+2)=9(3)=27Var(3X+2)=9(3)=27

  3. A constant shift has no effect on variance.

The requested value is 27.

Checks and common pitfalls: A constant shift has no effect on variance.

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

03 / Transfer#Worked example

A game pays t with probability 1/4 and 0 otherwise; entry fee is 2. Find expected net gain.

t=4t=4
  • Variance is nonnegative; variance of a sum includes covariance unless independence is justified.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
List the possible values and probabilities before computing moments.
Hint 2
Net gain=payment-2.
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    E(G)=4/4−2=−1E(G)=4/4-2=-1

  3. Expected net gain differs from the gross expected payment and is not a guaranteed outcome.

The requested value is -1.

Checks and common pitfalls: Expected net gain differs from the gross expected payment and is not a guaranteed outcome.

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

04 / Foundation#Your turn

Find E(X).

X=0,5;P=1/2,1/2X=0,5;\quad P=1/2,1/2
  • Variance is nonnegative; variance of a sum includes covariance unless independence is justified.
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.
E(X)=0/2+t/2E(X)=0/2+t/2
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    E(X)=5/2E(X)=5/2

  3. Expectation is a weighted average, not necessarily a possible outcome.

The requested value is 2.5.

Checks and common pitfalls: Expectation is a weighted average, not necessarily a possible outcome.

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

05 / Foundation#Your turn

Find E(X²).

X=0,6;P=1/2,1/2X=0,6;\quad P=1/2,1/2
  • Variance is nonnegative; variance of a sum includes covariance unless independence is justified.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
List the possible values and probabilities before computing moments.
Hint 2
Square the outcomes before averaging.
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    E(X2)=0/2+36/2E(X^2)=0/2+36/2

  3. Square outcomes before averaging; do not square the expectation instead.

The requested value is 18.

Checks and common pitfalls: Square outcomes before averaging; do not square the expectation instead.

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

06 / Foundation#Your turn

Find Var(X).

X=0,7;P=1/2,1/2X=0,7;\quad P=1/2,1/2
  • Variance is nonnegative; variance of a sum includes covariance unless independence is justified.
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.
Var(X)=E(X2)−E(X)2Var(X)=E(X^2)-E(X)^2
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    Var(X)=49/2−(7/2)2=49/4Var(X)=49/2-(7/2)^2=49/4

  3. The variance has squared outcome units.

The requested value is 12.25.

Checks and common pitfalls: The variance has squared outcome units.

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

07 / Foundation#Your turn

Find E(3X+2).

E(X)=8E(X)=8
  • Variance is nonnegative; variance of a sum includes covariance unless independence is justified.
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.
E(aX+b)=aE(X)+bE(aX+b)=aE(X)+b
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    E(3X+2)=3(8)+2=26E(3X+2)=3(8)+2=26

  3. Linearity of expectation does not require independence.

The requested value is 26.

Checks and common pitfalls: Linearity of expectation does not require independence.

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

08 / Standard#Your turn

Find E(3X+2).

E(X)=9E(X)=9
  • Variance is nonnegative; variance of a sum includes covariance unless independence is justified.
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.
E(aX+b)=aE(X)+bE(aX+b)=aE(X)+b
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    E(3X+2)=3(9)+2=29E(3X+2)=3(9)+2=29

  3. Linearity of expectation does not require independence.

The requested value is 29.

Checks and common pitfalls: Linearity of expectation does not require independence.

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

09 / Standard#Your turn

Find Var(3X+2).

Var(X)=10Var(X)=10
  • Variance is nonnegative; variance of a sum includes covariance unless independence is justified.
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.
Var(aX+b)=a2Var(X)Var(aX+b)=a^2Var(X)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    Var(3X+2)=9(10)=90Var(3X+2)=9(10)=90

  3. A constant shift has no effect on variance.

The requested value is 90.

Checks and common pitfalls: A constant shift has no effect on variance.

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

10 / Standard#Your turn

Independent X,Y have variances t and 2. Find Var(X+Y).

t=11t=11
  • Variance is nonnegative; variance of a sum includes covariance unless independence is justified.
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.
Var(X+Y)=Var(X)+Var(Y)Var(X+Y)=Var(X)+Var(Y)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    Var(X+Y)=11+2=13Var(X+Y)=11+2=13

  3. Independence supplies zero covariance here.

The requested value is 13.

Checks and common pitfalls: Independence supplies zero covariance here.

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

11 / Standard#Your turn

A game pays t with probability 1/4 and 0 otherwise; entry fee is 2. Find expected net gain.

t=12t=12
  • Variance is nonnegative; variance of a sum includes covariance unless independence is justified.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
List the possible values and probabilities before computing moments.
Hint 2
Net gain=payment-2.
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    E(G)=12/4−2=1E(G)=12/4-2=1

  3. Expected net gain differs from the gross expected payment and is not a guaranteed outcome.

The requested value is 1.

Checks and common pitfalls: Expected net gain differs from the gross expected payment and is not a guaranteed outcome.

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

12 / Transfer#Your turn

Independent X,Y have variances t and 2. Find Var(X+Y).

t=13t=13
  • Variance is nonnegative; variance of a sum includes covariance unless independence is justified.
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.
Var(X+Y)=Var(X)+Var(Y)Var(X+Y)=Var(X)+Var(Y)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    Var(X+Y)=13+2=15Var(X+Y)=13+2=15

  3. Independence supplies zero covariance here.

The requested value is 15.

Checks and common pitfalls: Independence supplies zero covariance here.

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

13 / Transfer#Your turn

A game pays t with probability 1/4 and 0 otherwise; entry fee is 2. Find expected net gain.

t=14t=14
  • Variance is nonnegative; variance of a sum includes covariance unless independence is justified.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
List the possible values and probabilities before computing moments.
Hint 2
Net gain=payment-2.
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    E(G)=14/4−2=1.5E(G)=14/4-2=1.5

  3. Expected net gain differs from the gross expected payment and is not a guaranteed outcome.

The requested value is 1.5.

Checks and common pitfalls: Expected net gain differs from the gross expected payment and is not a guaranteed outcome.

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

    • Compute expectation and variance and distinguish transformation from random independence.
    • Which condition is essential in numerical characteristics of random variables?
    • Does doubling all outcomes double or quadruple the variance?

    Board plan

    • Defining relation: Compute expectation and variance and distinguish transformation from random independence.
      E(X)=∑xpx;Var⁡(X)=E(X2)−[E(X)]2E(X)=\sum xp_x;\quad\operatorname{Var}(X)=E(X^2)-[E(X)]^2
    • Conditions: Variance is nonnegative; variance of a sum includes covariance unless independence is justified.

    Anticipated thinking

    • E(X²) is generally not [E(X)]².

    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 ↗