Statistics · Topic 4
Normal Distribution
X ~ N(μ, σ²)
P(a < X < b) via GDC
Standardise: Z = (X−μ)/σ
Symmetric bell curve. 68-95-99.7 rule applies.
💡 Key: Always state parameters μ and σ²
Statistics · Topic 4
Binomial Distribution
X ~ B(n, p)
P(X=r) = C(n,r)·pʳ·(1−p)ⁿ⁻ʳ
E(X) = np, Var(X) = np(1−p)
n independent trials, constant probability p.
💡 Key: Verify fixed n, independent, two outcomes
Statistics · Topic 4
Chi-squared Test
H₀: No association
df = (rows−1)(cols−1)
Expected = (row total × col total) / n
Tests independence between categorical variables.
💡 Key: All expected values ≥ 5
Statistics · Topic 4
Linear Regression
ŷ = ax + b (regression of y on x)
r = correlation coefficient
r² = coefficient of determination
Only use regression line within data range (interpolation).
💡 Key: r close to ±1 means strong linear correlation
Algebra · Topic 1
Arithmetic & Geometric Sequences
Arithmetic: uₙ = u₁ + (n−1)d
Sₙ = n/2·(2u₁ + (n−1)d)
Geometric: uₙ = u₁·rⁿ⁻¹, S∞ = u₁/(1−r)
Geometric S∞ exists only when |r| < 1.
💡 Key: Identify type from consecutive ratios or differences
Algebra · Topic 1
Matrices & Systems
det(A) = ad − bc
A⁻¹ = (1/det)·[[d,−b],[−c,a]]
AX = B ⟹ X = A⁻¹B
Unique solution exists iff det(A) ≠ 0.
💡 Key: Check determinant before finding inverse
Calculus & Models · Topic 2/5
Exponential Growth / Decay
P(t) = P₀·eᵏᵗ
k > 0: growth; k < 0: decay
Half-life: t₁/₂ = ln2 / |k|
Doubling time: t = ln2 / k
💡 Key: Solve for k using two known data points
Calculus & Models · Topic 5
Logistic Growth Model
P(t) = L / (1 + Ae^{−kt})
A = (L − P₀)/P₀
L = carrying capacity
Growth slows as population approaches L.
💡 Key: Inflection point at P = L/2
Calculus · Topic 5
Euler's Method (Numerical DE)
yₙ₊₁ = yₙ + h·f(xₙ, yₙ)
xₙ₊₁ = xₙ + h
Smaller h → greater accuracy
Approximates solutions to dy/dx = f(x,y).
💡 Key: Always show each step in the table
Calculus · Topic 5
Definite Integration
Area = ∫ₐᵇ |f(x)| dx
∫xⁿ dx = xⁿ⁺¹/(n+1) + C
∫eˣ dx = eˣ + C
Use |f(x)| when area is below x-axis.
💡 Key: Split integral at x-intercepts for area
Probability · Topic 4
Conditional Probability & Independence
P(A|B) = P(A∩B) / P(B)
Independent: P(A∩B) = P(A)·P(B)
Bayes: P(A|B) = P(B|A)·P(A)/P(B)
Tree diagrams: multiply along branches, add across paths.
💡 Key: Test independence: P(A|B) = P(A)?
Statistics · Topic 4
Poisson Distribution
X ~ Po(λ)
P(X=k) = e^{−λ}·λᵏ/k!
E(X) = Var(X) = λ
Models rare events in a fixed interval.
💡 Key: Mean equals variance — use to verify Poisson fit
Finance · Topic 1
Compound Interest & Finance
FV = PV·(1 + r/n)^{nt}
Continuous: FV = PV·eʳᵗ
Effective rate: (1 + r/n)ⁿ − 1
Use GDC Finance App (TVM) for complex problems.
💡 Key: Match n (periods) and r (rate per period)
Statistics · Topic 4
Hypothesis Testing
H₀: μ = μ₀ vs H₁: μ ≠ μ₀
t = (x̄ − μ₀) / (s/√n)
Reject H₀ if p-value < α
Type I error: reject true H₀ (probability = α).
💡 Key: State H₀ and H₁ clearly; conclude in context
Statistics · Topic 4
Spearman's Rank Correlation
rₛ = 1 − 6Σd²/[n(n²−1)]
d = difference in ranks
−1 ≤ rₛ ≤ 1
Non-parametric; used when data is not normally distributed.
💡 Key: Handle tied ranks by averaging