IIT M FOUNDATION FN EXAM FDF2 24 Dec 2023 - 24 Dec 2023

Concepts this paper tests

Auto-generated summary from the source site. Handy as a checklist; the lessons above explain each idea from scratch.

Here’s a structured summary of core topics, concepts, principles, and equations essential for the exam, followed by Mermaid knowledge graphs to visualize relationships.

Core Topics & Concepts

1. Probability & Statistics

A. Random Variables & Distributions

• Discrete Distributions:

  • Uniform (Discrete): PMF, CDF, Mean ((a+b)/2), Variance ((n²-1)/12).
  • Bernoulli: PMF, CDF, Mean (p), Variance (p(1-p)).
  • Binomial: PMF (nCk p^k (1-p)^(n-k)), Mean (np), Variance (np(1-p)).
  • Geometric: PMF ((1-p)^(k-1) p), Mean (1/p), Variance ((1-p)/p²).
  • Poisson: PMF ((e^-λ λ^k)/k!), Mean (λ), Variance (λ).

• Continuous Distributions:

  • Uniform (Continuous): PDF (1/(b-a)), CDF, Mean ((a+b)/2), Variance ((b-a)²/12).
  • Exponential: PDF (λe^-λx), CDF (1-e^-λx), Mean (1/λ), Variance (1/λ²).
  • Normal: PDF ((1/σ√2π) e^(-(x-μ)²/2σ²)), Mean (μ), Variance (σ²).
  • Gamma: PDF ((β^α/Γ(α)) x^(α-1) e^-βx), Mean (α/β), Variance (α/β²).
  • Beta: PDF (Γ(α+β)/Γ(α)Γ(β) x^(α-1) (1-x)^(β-1)), Mean (α/(α+β)), Variance (αβ/((α+β)²(α+β+1))).

B. Inequalities & Theorems

• Markov’s Inequality: P(X ≥ c) ≤ μ/c (for non-negative X).
• Chebyshev’s Inequality: P(|X-μ| ≥ kσ) ≤ 1/k².
• Weak Law of Large Numbers (WLLN): P(|X̄ - μ| > δ) ≤ σ²/(nδ²) where X̄ = (X₁ + ... + Xₙ)/n.
• Central Limit Theorem (CLT): (Y - nμ)/(σ√n) ≈ N(0,1) where Y = X₁ + ... + Xₙ.

C. Maximum Likelihood Estimation (MLE)

• Likelihood Function: L(θ) = ∏ f_X(x_i; θ).
• MLE Estimator: θ* = argmax_θ L(θ).

D. Joint & Conditional Distributions

• Joint PDF/PMF: f_X,Y(x,y).
• Conditional Expectation: E[X|Y=y] = ∫ x f_X|Y(x|y) dx.

2. Linear Algebra

A. Matrix Properties

• Positive Definite Matrices:

  • A is positive definite if xᵀAx > 0 for all x ≠ 0.

  • Properties:

    • A^k is positive definite for k ≥ 1.
    • Diagonal entries of A are positive.
    • A⁻¹ is positive definite.
    • rA is positive definite for r > 0.

• Eigenvalues & Eigenvectors:

  • Av = λv.
  • det(A - λI) = 0 (characteristic equation).
  • If A has n linearly independent eigenvectors, it is diagonalizable.

• Nullspace: Null(A) = {x | Ax = 0}.

B. Optimization & Convexity

• Convex Functions:

  • f(αx + (1-α)y) ≤ αf(x) + (1-α)f(y) for α ∈ [0,1].
  • Examples: f(x) = 1/x (convex for x > 0), g(x) = 1 + |x| (convex).

  • Composition rules:

    • g ∘ f is convex if g is convex and non-decreasing, and f is convex.
    • Sum of convex functions is convex.

3. Machine Learning & Optimization

A. Linear Regression

• Loss Function (MSE): L = (1/n) Σ (f(xᵢ) - yᵢ)² (for f(x) = w₁x₁ + w₂x₂ + ...). (Note: Question 3 uses L = - (1/n) Σ (f(xᵢ) - yᵢ)², which is negative MSE.)

B. Principal Component Analysis (PCA)

• Covariance Matrix: C = (1/n) XᵀX (for centered data).
• Projected Variance: Maximized along the principal component (eigenvector of C with largest eigenvalue).

C. Linear Programming

• Standard Form:

  • Maximize cᵀy subject to Ay ≤ b, y ≥ 0.

  • Example (Question 14):

    Maximize Z = 60y₁ + 45y₂
    Subject to:
      2y₁ + 2y₂ ≤ 200
      y₁ + 3y₂ ≤ 140
      y₁, y₂ ≥ 0

4. Calculus & Continuity

• Continuous Functions:

  • lim_{x→a} f(x) = f(a).

  • Examples:

    • sin(x)/x (continuous at x=0 if defined as 1).
    • Piecewise functions (e.g., x² + 3 for x ≥ 3, 3x + 3 for x < 3).

5. Geometry & Optimization

• Maximizing Area with Fixed Perimeter:

  • For a rectangle with perimeter P, the maximum area is achieved when it’s a square (side = P/4).
  • Example (Question 7): P = 600 ⇒ side = 150 ⇒ area = 22500.

• Constrained Optimization:

  • Lagrange Multipliers: For f(x,y) subject to g(x,y) = c, solve ∇f = λ∇g.
  • Example (Question 17): Maximize f(x,y) = x + y subject to x² + y² = 50. Solution: x = y = 5 ⇒ f(x,y) = 10.

Key Equations

Graph 2: Linear Algebra

Graph 3: Machine Learning

Graph 4: Calculus & Geometry