LEARN · EXPLAIN · REVISE

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

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

TOPIC 01

Numerical characteristics of random variables

Compute expectation and variance and distinguish transformation from random independence.

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

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

01 / Transfer#Your turn

t=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

List the possible values and probabilities before computing moments.
Apply the stated relation and retain its conditions.
$E(G)=14/4-2=1.5$
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.

Teacher preparation and assessment

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?

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