IIT M IMPROVEMENT FN EXAM FDI2 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 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, Variance.
  • 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)/p^2).
  • Poisson: PMF (e^(-λ) λ^k / k!), Mean & Variance (λ).

• Continuous Distributions:

  • Uniform (Continuous): PDF (1/(b-a)), CDF, Mean ((a+b)/2), Variance ((b-a)^2/12).
  • Exponential: PDF (λ e^(-λx)), CDF (1 - e^(-λx)), Mean (1/λ), Variance (1/λ^2).
  • Normal: PDF (1/(σ√(2π)) e^(-(x-μ)^2/(2σ^2))), Mean (μ), Variance (σ^2).
  • Gamma: PDF (β^α x^(α-1) e^(-βx) / Γ(α)), Mean (α/β), Variance (α/β^2).
  • 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/k^2.
• Weak Law of Large Numbers (WLLN): P(|X̄ - μ| > δ) ≤ σ^2 / (n δ^2).
• Central Limit Theorem (CLT): (Y - nμ) / (σ√n) ~ N(0,1) for Y = ΣX_i (i.i.d. X_i).
• Maximum Likelihood Estimation (MLE): θ* = argmax_θ Π f_X(x_i; θ).

C. Joint & Conditional Distributions

• Joint PDF/PMF: f_X,Y(x,y).
• Conditional Expectation: E[X|Y=y] = ∫ x f_X|Y(x|y) dx.
• Covariance Matrix: For data points X_i, compute mean, covariance, and eigenvalues for PCA.

2. Linear Algebra

A. Matrices & Properties

• Positive Definite Matrices:

  • A is positive definite if x^T A x > 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:

  • c is an eigenvalue of A iff (A - cI)x = 0 has a nontrivial solution.
  • If A has n linearly independent eigenvectors, it is diagonalizable.
• Nullspace: Solutions to A x = 0.

B. Optimization & Convexity

• Convex Functions:

  • 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.

• Linear Programming:

  • Maximize Z = 60y1 + 45y2 subject to constraints (e.g., machine/labor hours).

3. Machine Learning & Optimization

A. Linear Regression

• Loss Function (MSE): L = (1/n) Σ (f(x_i) - y_i)^2. Example: For f(x) = x1 + 2x2, compute L given data points.

B. Principal Component Analysis (PCA)

• Steps:

  1. Compute mean and covariance matrix C.
  2. Find eigenvalues and eigenvectors of C.
  3. Project data onto the eigenvector with the largest eigenvalue.
• Projected Variance: Largest eigenvalue of the covariance matrix.

C. Maximum Likelihood Estimation (MLE)

• For Exponential Distribution (Exp(λ)):

  • Likelihood: L(λ) = Π λ e^(-λ u_i).
  • MLE: λ* = 1 / (mean of samples).

4. Calculus & Continuity

• Continuous Functions:

  • sin(x)/x (continuous at x=0 if defined as 1).
  • Piecewise functions (check limits at boundaries).
  • Dirichlet function (discontinuous everywhere).

5. Geometry & Optimization

• Maximizing Area:

  • For a fixed perimeter, a square maximizes the area of a rectangle.
  • Example: Rope of length 600m → square with side 150m → area 22500 m².

• Constrained Optimization:

  • Maximize f(x,y) = x + y subject to x² + y² = 50 (use Lagrange multipliers or geometry).

Key Equations

2. Linear Algebra

3. Machine Learning

4. Optimization & Convexity

5. Calculus & Geometry