Statistics 2 and Maths 2 Revision Workbench

Statistics 2: Probability Engine Normal (Gaussian)

Mean (μ)

Std Dev (σ)

Lower Bound (a)

Upper Bound (b)

P(a ≤ X ≤ b) 0.6827

Z-Lower -1.00

Z-Upper +1.00

Variance (σ²) 100.00

Maths 2: Linear Algebra & Eigensystem (2×2) Characteristic Poly

Matrix A = [ [a, b], [c, d] ]

λ&sub1; (Eigenvalue 1) 1.38

λ&sub2; (Eigenvalue 2) 3.62

Trace Tr(A) 5.00

Determinant |A| 5.00

1. Characteristic Equation det(A - λI) = 0

λ² - (5.00)λ + 5.00 = 0

2. Eigenvectors Calculation

v&sub1; ≈ [-0.85, 0.53]⊤ | v&sub2; ≈ [0.53, 0.85]⊤

Revision Progress & Core Theorems 2 of 6 Completed

Track essential topics for the end-term exams in Statistics 2 (IITM BS DS) and Maths 2.

Central Limit Theorem (CLT) Stats 2
As sample size n → ∞, the sampling distribution of sample mean ‾X converges to Normal(μ, σ²/n).
Z = (X̄ - μ) / (σ / √n) ∼ N(0, 1)
Matrix Diagonalization & Spectral Theorem Maths 2
If A has n linearly independent eigenvectors, A = P D P⁻¹ where D contains eigenvalues.
A v = λ v ⇒ A = P diag(λ&sub1;, λ&sub2;) P⁻¹
Hypothesis Testing & p-values Stats 2
Null hypothesis H₀ vs H₁, Type I error α, Type II error β, p-value ≤ α rejects H₀.
Type I error = P(Reject H₀ | H₀ is True) = α
Singular Value Decomposition (SVD) Maths 2
Any m×n matrix can be factored as U Σ V⊤, where Σ holds singular values σ&sub1; ≥ σ&sub2; ≥ ...
A = U Σ V⊤ where σᵢ = √λᵢ(A⊤A)
Maximum Likelihood Estimation (MLE) Stats 2
Find parameter θ maximizing log-likelihood L(θ; x) = ∑ log f(xᵢ; θ). Set ∂ln L/∂θ = 0.
θ̂_{MLE} = argmax ∑ ln f(xᵢ; θ)
Gram-Schmidt & QR Decomposition Maths 2
Orthogonalization of column vectors: A = Q R where Q has orthonormal columns and R is upper triangular.
u₂ = v₂ - proj_{u₁}(v₂); A = Q R

Personal Revision Notes Persistent