FAD1015 Final Exam Scope — Complete Guide

[!note] Single authoritative exam scope document. Merges multiple data sources:

  1. Past year patterns — 2 transcribed finals (2023-24, 2024-25)
  2. Student leaks — WhatsApp messages from Adian Sani, Anthonny (2026-05-14)
  3. Tutorial teacher confirmation — YX (matrix teacher) confirmed topics

1. Exam Structure

Confirmed Format (2025-2026)

Section Format Marks Time Allocation
Part A 2 questions × multi-part ~40 ~45 min
Part B 4 questions × multi-part ~60 ~75 min
Total 6 questions ~100 ~2 hours

Part A Breakdown

Question Topic Sub-parts Priority
Q1 Binomial Distribution 6 parts ⭐⭐⭐
Q2 Poisson + Uniform/Exponential + Matrices 4 parts ⭐⭐⭐

Part B Breakdown

Question Topic Sub-parts Priority
Q3 CDF + Poisson Calculations 2 parts ⭐⭐⭐
Q4 Continuous/Discrete + Probability 4 parts ⭐⭐⭐
Q5 Hypothesis Testing 5 parts ⭐⭐⭐
Q6 Matrices in R 5 parts ⭐⭐⭐⭐

2. Part A — Detailed Analysis

Q1: Binomial Distribution (6 Sub-parts)

From Adian Sani (confirmed by YX tutorial teacher):

Part Topic What to Know
(a) Definition "X follows a binomial distribution if..." — all 4 conditions
(b) Identify binomial Given scenario, determine if binomial applies
(c) Parameters Identify n (trials) and p (probability of success)
(d) Statistics vs Parameters Distinguish sample statistics from population parameters
(e) Characteristics Mean = np, Variance = np(1-p), SD = √(np(1-p))
(f) Type of distribution Discrete vs continuous identification

Binomial Definition (Memorize):

X ~ B(n, p) if:

  1. Fixed number of trials (n)
  2. Each trial has two outcomes (success/failure)
  3. Constant probability of success (p)
  4. Trials are independent

Key Formulas:

  • P(X = r) = C(n,r) × p^r × (1-p)^(n-r)
  • E(X) = np
  • Var(X) = np(1-p)

Q2: Poisson + Uniform/Exponential + Matrices (4 Sub-parts)

From Adian Sani:

Part Topic Key Concepts
(a) Poisson Approximation When to use: n > 20, p < 0.05 (or np < 10)
(b) Conversion Law λ = np
(c) Uniform/Exponential PDF, CDF, mean, variance formulas
(d) Matrices Refer to Tutorial 13 Q1

Poisson Approximation Rules:

  • Use when n is large and p is small
  • λ = np (same mean as binomial)
  • P(X = r) = (e^(-λ) × λ^r) / r!

Uniform Distribution:

  • X ~ U(a, b)
  • f(x) = 1/(b-a) for a ≤ x ≤ b
  • E(X) = (a+b)/2
  • Var(X) = (b-a)²/12

Exponential Distribution:

  • X ~ Exp(λ)
  • f(x) = λe^(-λx) for x ≥ 0
  • E(X) = 1/λ
  • Var(X) = 1/λ²
  • P(X > x) = e^(-λx)

3. Part B — Detailed Analysis

Q3: Cumulative Distribution + Poisson Calculations (2 Sub-parts)

Part Topic Skills Required
(a) CDF Calculate F(x) = P(X ≤ x)
(b) Poisson Calculations PMF and cumulative probabilities

CDF Skills:

  • Build CDF table for discrete distributions
  • Calculate cumulative probabilities
  • F(x) = Σ P(X = k) for all k ≤ x

Poisson Calculations:

  • Individual probability: P(X = r)
  • Cumulative: P(X ≤ r)
  • Right-tail: P(X ≥ r) = 1 - P(X ≤ r-1)
  • Range: P(a ≤ X ≤ b) = P(X ≤ b) - P(X ≤ a-1)

Q4: Continuous/Discrete + Probability (4 Sub-parts)

Part Topic Skills Required
(a) Identify distribution type Continuous vs discrete
(b) Smallest/Biggest n Find range boundaries
(c) P(a < X < b) Interval probability
(d) Mean and SD E(X) and σ calculations

Distribution Identification:

  • Discrete: Binomial, Poisson, Geometric
  • Continuous: Normal, Uniform, Exponential

Interval Probability:

  • Continuous: P(a < X < b) = ∫[a to b] f(x) dx
  • Discrete: P(a < X < b) = Σ P(X = k) for a < k < b

Q5: Hypothesis Testing (5 Sub-parts) ⭐ CRITICAL

From Adian Sani:

Step Action Details
1 Find critical value Z-table (σ known) or t-table (σ unknown)
2 Find p-value From test statistic
3 Compare with α Usually α = 0.05
4 Make conclusion Reject or fail to reject H₀
5 Confidence interval Construct CI, then conclude

Hypothesis Testing Procedure:

  1. State hypotheses:

    • H₀: μ = μ₀ (null)
    • H₁: μ ≠ μ₀ (two-tailed) or μ > μ₀ or μ < μ₀ (one-tailed)

  2. Choose test:

    • Z-test: σ known OR n > 30
    • T-test: σ unknown AND n < 30

  3. Calculate test statistic:

    • Z = (x̄ - μ₀) / (σ/√n)
    • t = (x̄ - μ₀) / (s/√n)

  4. Find critical value / p-value

  5. Decision rule:

    • Reject H₀ if |test stat| > critical value
    • Reject H₀ if p-value < α

  6. Conclusion:

    • If reject: "There is sufficient evidence at α = 0.05 level to conclude that..."
    • If fail to reject: "There is insufficient evidence..."

Confidence Interval:

  • x̄ ± Z_(α/2) × (σ/√n) — Z-interval
  • x̄ ± t_(α/2,n-1) × (s/√n) — T-interval

[!warning] Hypothesis Testing in R NOT Tested Chen Jing confirmed: Hypothesis testing IN R is NOT coming out. All hypothesis testing is by-hand only — Z-table, t-table, formulas. No t.test() R code will appear. See FAD1015 Exam Leaks 2025-2026#Q5 Hypothesis Testing for details.

Q6: Matrices in R (5 Sub-parts) ⭐⭐⭐⭐ HIGHEST VOLUME

From Adian Sani:

Part Topic R Skills
(a) Matrix inverse and transpose solve(), t()
(b) Cramer's rule Calculate determinants, substitute columns
(c) R output prediction Read code, predict result
(d) Error detection Find mistakes in R code
(e) Scatterplot + Descriptive summary plot(), summary()

[!warning] Most Questions in R The tutorial teacher (YX) confirmed: Most matrix questions will be in R programming format.

Essential R Commands:

Matrix Creation:

A <- matrix(c(1,2,3,4), nrow=2, ncol=2)     # by column
cbind(v1, v2)                               # column bind
rbind(v1, v2)                               # row bind
diag(n)                                     # identity matrix

Matrix Operations:

t(A)              # transpose
solve(A)          # inverse
det(A)            # determinant
A %*% B           # matrix multiplication
A * B             # element-wise multiplication
A + B             # addition
A - B             # subtraction

Solving Systems:

solve(A, b)       # solve Ax = b

Descriptive Statistics:

mean(x)
sd(x)
var(x)
median(x)
summary(x)        # min, Q1, median, mean, Q3, max

Plotting:

plot(x, y)                    # scatterplot
plot(x, y, main="Title")      # with title
plot(x, y, xlab="X", ylab="Y") # with axis labels
hist(x)                       # histogram
boxplot(x)                    # box plot

Cramer's Rule (by hand): For system Ax = b:

  1. Calculate D = det(A)
  2. For x₁: Replace column 1 with b → A₁, calculate D₁ = det(A₁)
  3. For x₂: Replace column 2 with b → A₂, calculate D₂ = det(A₂)
  4. x₁ = D₁/D, x₂ = D₂/D

4. General Exam Tips

Normal Distribution Questions

From Anthonny's Maths Tips:

  • Questions give many numbers
  • Case study in long sentences
  • Irrelevant information included
  • Don't be tricked — identify what's actually needed

Strategy:

  1. Read the entire question
  2. Identify what distribution is being asked
  3. Extract only the relevant numbers
  4. Apply correct formula

Statistical Analysis

  • Students struggle with statistical analysis part
  • Check tutorial questions for practice

Matrices

  • Matrix questions are easy conceptually
  • R programming format is the challenge
  • Practice R syntax extensively

5. Key Topics Summary by Priority

Priority Topic Question Why Critical
⭐⭐⭐⭐ Matrices in R Q6 Most marks, highest volume
⭐⭐⭐⭐ Hypothesis Testing Q5 Multi-step, easy to lose marks
⭐⭐⭐ Binomial Definition Q1 Must be precise
⭐⭐⭐ Poisson Approximation Q2 Common calculation error
⭐⭐⭐ CDF Calculations Q3 Builds on earlier concepts
⭐⭐⭐ Probability Intervals Q4 Application of distributions

6. Tutorial References

Tutorial Question Topic Priority
Tuto 5 All CRV — Continuous Random Variables ⭐⭐⭐
Tuto 6 All CRV — Mean, Variance, Probability ⭐⭐⭐
Tuto 13 Q1 Matrices ⭐⭐⭐⭐
Tuto 14 All Matrices in R ⭐⭐⭐⭐

7. Formula Quick Reference

Binomial Distribution

Formula Purpose
P(X = r) = C(n,r) p^r (1-p)^(n-r) Probability of exactly r successes
E(X) = np Expected value
Var(X) = np(1-p) Variance
SD(X) = √(np(1-p)) Standard deviation

Poisson Distribution

Formula Purpose
P(X = r) = (e^(-λ) λ^r) / r! Probability of r events
E(X) = λ Expected value
Var(X) = λ Variance
λ = np (approximation) Convert from binomial

Uniform Distribution

Formula Purpose
f(x) = 1/(b-a) PDF
F(x) = (x-a)/(b-a) CDF
E(X) = (a+b)/2 Mean
Var(X) = (b-a)²/12 Variance

Exponential Distribution

Formula Purpose
f(x) = λe^(-λx) PDF
F(x) = 1 - e^(-λx) CDF
P(X > x) = e^(-λx) Survival function
E(X) = 1/λ Mean
Var(X) = 1/λ² Variance

Normal Distribution

Formula Purpose
Z = (X - μ) / σ Standardize
P(X < x) = P(Z < (x-μ)/σ) Probability calculation

Hypothesis Testing

Formula Purpose
Z = (x̄ - μ₀) / (σ/√n) Z-test statistic
t = (x̄ - μ₀) / (s/√n) T-test statistic
CI = x̄ ± Z_(α/2) × (σ/√n) Confidence interval

Matrices

Formula Purpose
det(A) for 2×2: ad - bc Determinant
A⁻¹ = (1/det(A)) × adj(A) Inverse
x = A⁻¹b Solve system
Cramer's: xᵢ = det(Aᵢ) / det(A) Cramer's rule

8. Pre-Exam Checklist

Part A Preparation

  • [ ] Binomial definition — all 4 conditions memorized
  • [ ] Can identify if scenario is binomial
  • [ ] Parameters n and p identification
  • [ ] Statistics vs parameters distinction
  • [ ] Poisson approximation conditions (n > 20, p < 0.05)
  • [ ] λ = np conversion
  • [ ] Uniform distribution PDF, mean, variance
  • [ ] Exponential distribution PDF, mean, variance
  • [ ] Tutorial 13 Q1 — matrix problem solved

Part B Preparation

  • [ ] CDF calculations for all distributions
  • [ ] Poisson individual and cumulative probabilities
  • [ ] Identify continuous vs discrete distributions
  • [ ] Find smallest/biggest n values
  • [ ] Calculate P(a < X < b) for all distributions
  • [ ] Mean and SD formulas for all distributions

Hypothesis Testing (Critical)

  • [ ] State H₀ and H₁ correctly
  • [ ] Choose Z-test vs T-test
  • [ ] Calculate test statistic
  • [ ] Find critical values (Z-table, t-table)
  • [ ] Calculate p-values
  • [ ] Compare p-value with α
  • [ ] Write proper conclusions
  • [ ] Construct confidence intervals

Matrices in R (Most Critical)

  • [ ] Create matrices with matrix(), cbind(), rbind()
  • [ ] Transpose with t()
  • [ ] Inverse with solve()
  • [ ] Determinant with det()
  • [ ] Matrix multiplication with %*%
  • [ ] Solve systems with solve(A, b)
  • [ ] Cramer's rule by hand
  • [ ] Predict R code output
  • [ ] Detect errors in R code
  • [ ] plot() for scatterplots
  • [ ] summary() for descriptive stats

General

  • [ ] Normal distribution long questions — filter irrelevant info
  • [ ] Tutorial 5, 6, 13, 14 completed
  • [ ] Past year papers reviewed
  • [ ] Calculator with statistical functions
  • [ ] Z-table and t-table ready

9. Common Mistakes to Avoid

Mistake Why It Happens Prevention
Using Z-test when σ unknown Forgetting t-test conditions Check: σ known? n > 30?
Wrong alternative hypothesis Misreading "greater than" vs "less than" Underline key words
Forgetting continuity correction Binomial → Normal Add/subtract 0.5
R syntax errors Confusing %*% vs * Practice R code
Wrong confidence level α = 0.05 vs α = 0.01 Double-check question
Not identifying distribution Long case studies Extract key words first

10. Time Allocation Strategy

Section Time Per Question
Part A 45 min Q1: 25 min, Q2: 20 min
Part B 75 min ~18-20 min each
Review 10 min Check calculations

Recommended Order:

  1. Q1 (Binomial) — straightforward, builds confidence
  2. Q6 (Matrices in R) — highest marks, do while fresh
  3. Q5 (Hypothesis Testing) — methodical, needs focus
  4. Q3 (CDF) — calculation-based
  5. Q4 (Probability) — apply concepts
  6. Q2 (Poisson/Uniform) — finish strong

[!tip] Key Success Factors

  1. R syntax practice — Most marks depend on R programming
  2. Hypothesis testing steps — Follow the 5-step procedure exactly
  3. Tutorial 13 Q1 — Master this matrix problem
  4. Normal distribution filtering — Ignore irrelevant numbers
  5. Confidence intervals — Practice construction and interpretation

11. Study Guide — What to Read & What to Master

Part A Topics — Priority Lectures

Q1: Binomial Distribution (6 Sub-parts)

Read: FAD1015 L13 — Binomial Distribution

Must Master:

  • Definition (Part a): Memorize all 4 conditions:
    1. Fixed number of trials (n)
    2. Two possible outcomes (success/failure)
    3. Independent trials
    4. Constant probability of success (p)
  • Identify binomial (Part b): Apply the 4-condition check
  • Parameters (Part c): Identify n and p from problem context
  • Statistics vs parameters (Part d): Sample mean x̄ vs population mean μ
  • Characteristics (Part e):
    • Mean: μ = np
    • Variance: σ² = npq
    • Standard deviation: σ = √(npq)
  • Distribution type (Part f): Discrete vs continuous identification

Key Formulas: $$P(X = x) = \binom{n}{x} p^x q^{n-x}$$

Using Tables:

  • P(X ≥ x): Read directly
  • P(X ≤ x): 1 − P(X ≥ x+1)
  • P(X = x): P(X ≥ x) − P(X ≥ x+1)

When p > 0.5: Use complementary probability with q = 1−p

Q2: Poisson + Uniform/Exponential + Matrices

Read: FAD1015 L14 — Poisson Distribution, FAD1015 L17-L18 — Uniform & Exponential Distributions + R Intro

Poisson Approximation (Part a-b):

  • Conditions: n > 20 AND p < 0.05 (or np < 10)
  • λ = np
  • P(X = x) = (λˣe⁻ˣ) / x!
  • Mean = Variance = λ

Uniform Distribution (Part c):

  • PDF: f(x) = 1/(b−a) for a ≤ x ≤ b
  • Mean: E[X] = (a+b)/2
  • Variance: Var(X) = (b−a)²/12
  • P(c < X < d) = (d−c)/(b−a)

Exponential Distribution (Part c):

  • PDF: f(x) = λe⁻ˣˣ for x ≥ 0
  • CDF: F(x) = 1 − e⁻ˣˣ
  • Mean: E[X] = 1/λ
  • Standard deviation: σ = 1/λ
  • Memoryless property: P(X > s+t | X > s) = P(X > t)

Matrices (Part d):

  • Tutorial 13 Q1 — master this specific problem
  • Matrix types: row, column, square, diagonal, identity
  • Matrix operations: addition, scalar multiplication, matrix multiplication
  • Transpose: (AB)ᵀ = BᵀAᵀ

Part B Topics — Priority Lectures

Q3: CDF + Poisson Calculations

Read: FAD1015 Week 4 — Discrete Random Variables (PDF & CDF), FAD1015 Week 6 — Continuous Random Variables

Must Master:

  • CDF definition: F(x) = P(X ≤ x)
  • Relationship between PDF and CDF
  • Calculating probabilities for discrete distributions:
    • P(X ≤ k) = Σ P(X = i) for i = 0 to k
    • P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a−1)
  • Calculating probabilities for continuous distributions:
    • P(a < X < b) = F(b) − F(a)
  • Poisson cumulative probabilities using tables

Q4: Continuous/Discrete + Probability + Mean/SD

Read: FAD1015 Week 5 — Mean & Variance (Discrete & Continuous), FAD1015 L15-L16 — Normal Distribution & Approximation

Must Master:

  • Identifying distribution types:
    • Discrete: Binomial, Poisson, Geometric
    • Continuous: Normal, Uniform, Exponential
  • Finding smallest/largest n values
  • Interval probability calculations
  • Mean and variance formulas for all distributions

Normal Distribution (for filtering long questions):

  • Standardization: Z = (X − μ) / σ
  • Using Z-tables to find probabilities
  • Key skill: Filter irrelevant information in long case studies

Q5: Hypothesis Testing ⭐ CRITICAL (5 Sub-parts)

Read: FAD1015 L23-L24 — Hypothesis Testing About the Mean, FAD1015 L25-L26 — Hypothesis Testing in R

[!warning] By-hand only — no R Hypothesis testing in R is confirmed NOT tested. All Q5 work is by-hand using Z/tables. No t.test() or R code.

Must Master All 5 Steps:

Step 1: State hypotheses

  • H₀: μ = μ₀ (null)
  • H₁: μ ≠ μ₀ (two-tailed) or μ > μ₀ (right-tailed) or μ < μ₀ (left-tailed)

Step 2: Select α

  • Usually α = 0.05
  • For Part 5: Find critical value from Z-table or t-table

Step 3: Calculate test statistic

  • Z-test (σ known or n ≥ 30): Z = (x̄ − μ₀) / (σ/√n)
  • T-test (σ unknown and n < 30): t = (x̄ − μ₀) / (s/√n), df = n−1

Step 4: Decision methods

  • Traditional: Compare test statistic to critical value
  • P-value: Find P(Z > |z|) from tables
  • Confidence Interval: Check if μ₀ falls inside CI

Step 5: Conclusion

  • If reject H₀: "There is sufficient evidence at α = 0.05 to conclude that..."
  • If fail to reject: "There is insufficient evidence..."

Critical Values to Memorize:

Test α = 0.05 α = 0.01
Two-tailed Z ±1.96 ±2.576
Right-tailed Z 1.645 2.326
Left-tailed Z −1.645 −2.326

Q6: Matrices in R ⭐⭐⭐⭐ HIGHEST VOLUME (5 Sub-parts)

Read: FAD1015 L27-L28 — Matrices (Types & Operations), FAD1015 L29-L30 — Matrices (Inverse & Systems of Equations), FAD1015 L19 — Input Data & Descriptive Statistics in R

Must Master — R Programming:

Matrix Creation:

A <- matrix(c(1,2,3,4), nrow=2, ncol=2)  # by column
cbind(v1, v2)                            # column bind
rbind(v1, v2)                            # row bind
diag(n)                                  # identity matrix

Matrix Operations:

t(A)              # transpose
solve(A)          # inverse
det(A)            # determinant
A %*% B           # matrix multiplication (NOT A*B)

Solving Systems:

solve(A, b)       # Solve Ax = b

Cramer's Rule (by hand): For system Ax = b:

  1. D = det(A)
  2. D₁ = det(A with column 1 replaced by b)
  3. D₂ = det(A with column 2 replaced by b)
  4. x₁ = D₁/D, x₂ = D₂/D

Descriptive Statistics in R:

mean(x)
sd(x)
var(x)
median(x)
summary(x)        # min, Q1, median, mean, Q3, max

Plotting:

plot(x, y)                    # scatterplot
plot(x, y, main="Title")      # with title
hist(x)                       # histogram
boxplot(x)                    # box plot

Key Distinction:

  • %*% = matrix multiplication
  • * = element-wise multiplication

Essential R Commands Summary

Task R Command
Create matrix matrix(data, nrow, ncol)
Column bind cbind(v1, v2)
Row bind rbind(v1, v2)
Transpose t(A)
Inverse solve(A)
Determinant det(A)
Matrix multiply A %*% B
Solve Ax=b solve(A, b)
Mean mean(x)
Standard deviation sd(x)
Variance var(x)
Summary stats summary(x)
Scatterplot plot(x, y)

Tutorial Study Priority

Tutorial Priority Focus
Tutorial 5 ⭐⭐⭐ CRV — Continuous Random Variables
Tutorial 6 ⭐⭐⭐ Mean, Variance, Probability
Tutorial 13 ⭐⭐⭐⭐ Matrices — Q1 specifically
Tutorial 14 ⭐⭐⭐⭐ Matrices in R

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