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Random variables and distributions: mixed review

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

TOPIC 01

Random variables and distributions: mixed review

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

Connect the chapter skills without relying on the order of the exercises.

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

Find P(A|B) from the given probabilities.

P(A\cap B)=1/24,\quad P(B)=2/24

The conditioning event must have positive probability; a total-probability partition is exhaustive and disjoint.

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.

P(A|B)=P(A\cap B)/P(B)

Worked solution

Express the event with a conditional probability or a disjoint partition.
Apply the stated relation and retain its conditions.
$P(A|B)=(1/u)/(2/u)=1/2$
Restrict the sample space to B.

The requested value is 0.5.

Checks and common pitfalls: Restrict the sample space to B.

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

02 / Foundation#Your turn

Find P(X≥t).

X\in\{0,21,22\};\quad P=(1/4,1/4,1/2)

Every probability lies in [0,1]; values and events must not be double-counted.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

List the possible values and probabilities before computing moments.

Hint 2

Add probabilities at t and t+1.

Worked solution

List the possible values and probabilities before computing moments.
Apply the stated relation and retain its conditions.
$P(X\ge t)=1/4+1/2=3/4$
The threshold selects values, then their probabilities are added.

The requested value is 0.75.

Checks and common pitfalls: The threshold selects values, then their probabilities are added.

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

03 / Foundation#Your turn

Find Var(X).

X=0,22;\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(X^2)-E(X)^2

Worked solution

List the possible values and probabilities before computing moments.
Apply the stated relation and retain its conditions.
$Var(X)=484/2-(22/2)^2=484/4$
The variance has squared outcome units.

The requested value is 121.

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

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

04 / Foundation#Your turn

X~Bin(m,1/4). Find Var(X).

m=4

Binomial trials have fixed n, common p and independence; hypergeometric sampling uses a finite population without replacement.

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)=mp(1-p)

Worked solution

List the possible values and probabilities before computing moments.
Apply the stated relation and retain its conditions.
$Var(X)=4(1/4)(3/4)=12/16$
Include the failure probability as well as the success probability.

The requested value is 0.75.

Checks and common pitfalls: Include the failure probability as well as the success probability.

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

05 / Foundation#Your turn

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

t=24

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.

\sigma=\sqrt{\sigma^2}

Worked solution

List the possible values and probabilities before computing moments.
Apply the stated relation and retain its conditions.
$\sigma=\sqrt4=2$
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.

06 / Foundation#Your turn

From a bag of t red and 2 blue balls, two are drawn without replacement. Given the first is red, find the probability the second is blue.

t=25

The conditioning event must have positive probability; a total-probability partition is exhaustive and disjoint.

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

After a red draw, total=t+1 and blue=2.

Worked solution

Express the event with a conditional probability or a disjoint partition.
Apply the stated relation and retain its conditions.
$P=2/26$
The color count and denominator must reflect the information already given.

The requested value is 0.0769230769231.

Checks and common pitfalls: The color count and denominator must reflect the information already given.

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

07 / Standard#Your turn

Give the distribution of Y=2X+1.

X=0,26;\quad P=1/3,2/3

Every probability lies in [0,1]; values and events must not be double-counted.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

List the possible values and probabilities before computing moments.

Hint 2

Map each support value through Y=2X+1.

Worked solution

List the possible values and probabilities before computing moments.
Apply the stated relation and retain its conditions.
$0\mapsto1,\quad 26\mapsto53$
An injective value transformation keeps the corresponding probabilities.

The requested relation or conclusion is shown below.

Y=1,53;\quad P=1/3,2/3

Checks and common pitfalls: An injective value transformation keeps the corresponding probabilities.

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.

08 / Standard#Your turn

Find Var(3X+2).

Var(X)=27

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)=a^2Var(X)

Worked solution

List the possible values and probabilities before computing moments.
Apply the stated relation and retain its conditions.
$Var(3X+2)=9(27)=243$
A constant shift has no effect on variance.

The requested value is 243.

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

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

09 / Standard#Your turn

From t red and 3 blue balls, select 2 without replacement. Find the probability exactly one is red.

t=28

Binomial trials have fixed n, common p and independence; hypergeometric sampling uses a finite population without replacement.

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.

C_t^1C_3^1/C_{t+3}^2

Worked solution

Express the event with a conditional probability or a disjoint partition.
Apply the stated relation and retain its conditions.
$P=84/465$
Choose one from each color; do not add a spurious ordering factor.

The requested value is 0.18064516129.

Checks and common pitfalls: Choose one from each color; do not add a spurious ordering factor.

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

10 / Standard#Your turn

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

t=29

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.

\Phi(-1)=1-\Phi(1)

Worked solution

Express the event with a conditional probability or a disjoint partition.
Apply the stated relation and retain its conditions.
$P(-1\le Z\le1)=2\Phi(1)-1=0.6826$
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.

11 / Standard#Your turn

Choose machine 1 with probability 1/(t+2), otherwise machine 2; their failure rates are 1/10 and 1/5. Given failure, find P(machine 1).

t=30

The conditioning event must have positive probability; a total-probability partition is exhaustive and disjoint.

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.

P(M_1|F)=P(M_1\cap F)/P(F)

Worked solution

Express the event with a conditional probability or a disjoint partition.
Apply the stated relation and retain its conditions.
$P(F)=63/[10(32)]$
Apply the stated relation and retain its conditions.
$P(M_1|F)=\frac{1/[10(32)]}{63/[10(32)]}=1/63$
The posterior denominator includes failures from both machines.

The requested value is 0.015873015873.

Checks and common pitfalls: The posterior denominator includes failures from both machines.

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

12 / Standard#Your turn

Can these numbers be a probability distribution? Explain.

X=0,31,32;\quad P=0.6,0.5,-0.1

Every probability lies in [0,1]; values and events must not be double-counted.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

List the possible values and probabilities before computing moments.

Hint 2

Check nonnegativity as well as the total.

Worked solution

List the possible values and probabilities before computing moments.
Apply the stated relation and retain its conditions.
$0.6+0.5-0.1=1$
Apply the stated relation and retain its conditions.
$P(X=t+1)=-0.1<0$
A sum of one alone is insufficient.

The requested relation or conclusion is shown below.

\text{invalid}

Checks and common pitfalls: A sum of one alone is insufficient.

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.

13 / Transfer#Your turn

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

t=32

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)

Worked solution

List the possible values and probabilities before computing moments.
Apply the stated relation and retain its conditions.
$Var(X+Y)=32+2=34$
Independence supplies zero covariance here.

The requested value is 34.

Checks and common pitfalls: Independence supplies zero covariance here.

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

14 / Transfer#Your turn

Independent trials succeed with probability 1/2. Find the least n for success at least once with probability ≥1−2^(−m).

m=3

Binomial trials have fixed n, common p and independence; hypergeometric sampling uses a finite population without replacement.

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.

1-2^{-n}\ge1-2^{-m}

Worked solution

Express the event with a conditional probability or a disjoint partition.
Apply the stated relation and retain its conditions.
$2^{-n}\le2^{-3}\Rightarrow n\ge3$
Use the complementary event of no successes.

The requested value is 3.

Checks and common pitfalls: Use the complementary event of no successes.

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

15 / Transfer#Your turn

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

t=34

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

Express the event with a conditional probability or a disjoint partition.
Apply the stated relation and retain its conditions.
$P(X=\mu)=0$
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.

End-of-lesson check and correction

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Teacher preparation and assessment

Question sequence

Ask students to name the relevant condition before calculating.

Board plan

Compare valid methods and annotate their conditions.

Anticipated thinking

A correct final value may still hide a missing assumption.

Assessment checklist

Check the method, conditions, reasoning and interpretation separately.