IIT M FOUNDATION DIPLOMA FN EXAM - 22 Dec 2024

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 this exam, followed by Mermaid knowledge graphs to visualize relationships.

Core Topics & Key Concepts

1. Probability & Random Variables

• Distributions:

  • Normal Distribution: $X \sim N(\mu, \sigma^2)$.

    • Linear combinations: If $X \sim N(\mu_X, \sigma_X^2)$ and $Y \sim N(\mu_Y, \sigma_Y^2)$, then $aX + bY \sim N(a\mu_X + b\mu_Y, a^2\sigma_X^2 + b^2\sigma_Y^2)$.
    • Key Equation:

• Uniform Distribution: $X \sim \text{Uniform}(a, b)$.

  • CDF of $\min(X, Y)$: For independent $X, Y \sim \text{Uniform}(0,1)$,

• Exponential Distribution: Joint PDF $f_{X,Y}(x,y) = e^{-(x+y)}$ for $x, y \geq 0$.

  • Transformation of Variables: For $U = X + Y$, $V = X - Y$, the joint PDF is:

• Multivariate Normal Distribution:

  • $X \sim N(\mu, \Sigma)$. For $Y = a^T X$, $Y \sim N(a^T\mu, a^T\Sigma a)$.
  • Key Equation:

2. Linear Algebra

• Matrix Properties:

  • Positive Definite: $A$ is positive definite if $x^T A x > 0$ for all $x \neq 0$.
  • Symmetric Matrix: $A = A^T$.
  • Quadratic Forms: $f(x) = x^T A x$. For $f(x) = x_1^2 + 2x_1x_2 + 5x_2^2 + 3x_3^2$, $A$ is:

(Symmetric and positive definite.)

• Eigenvalues/Eigenvectors:

  • Unitary Matrix: $A^* A = I$. For complex matrices, check $A^* = A^{-1}$.
  • Hermitian Matrix: $A = A^*$. Eigenvalues are real; eigenvectors are orthogonal.

  • Key Facts:

    • Non-zero eigenvalues of $A^T A$ and $AA^T$ are identical.
    • $\det(A) \neq 0$ implies nullity $= 0$ (no zero eigenvalues).
    • Full-rank $A$ has $\text{rank}(A) = n$.

3. Convexity & Optimization

• Convex Sets:

  • A set $S$ is convex if $\lambda a + (1-\lambda)b \in S$ for all $a, b \in S$, $\lambda \in [0,1]$.
  • Examples: Midpoint $(0.5a + 0.5b)$ and $0.7a + 0.3b$ are in $S$.

• Convex Functions:

  • $f$ is convex if $f(\lambda x + (1-\lambda)y) \leq \lambda f(x) + (1-\lambda)f(y)$.

  • Examples:

    • $f(x) = x \ln x$ (convex on $[0, \infty)$).
    • $f(x,y) = e^{2x + y}$ (convex, as Hessian is positive semidefinite).
    • $f(x,y) = -\ln(xy)$ (convex on $(0, \infty)^2$).
  • Composition: If $f$ is convex and non-decreasing, and $g$ is convex, then $f \circ g$ is convex.

• Optimization Algorithms:

  • Gradient Descent: Update rule:

Example: For $f(x,y) = x^2 + y^2 -6x -8y$, $\nabla f = (2x -6, 2y -8)$. Starting at $(1,2)$, first iteration:

$10x + 10y = 46$.

• KKT Conditions:

  • For constrained optimization $\min f(x)$ s.t. $g_i(x) \leq 0$, KKT requires:

    1. Stationarity: $\nabla f(x) + \sum \lambda_i \nabla g_i(x) = 0$.
    2. Primal Feasibility: $g_i(x) \leq 0$.
    3. Dual Feasibility: $\lambda_i \geq 0$.
    4. Complementary Slackness: $\lambda_i g_i(x) = 0$.
  • Example: For $f(x_1, x_2) = x_1^2 + x_2^2$ with $g_1 = x_1 + x_2 -1 \leq 0$, $g_2 = -x_1 \leq 0$, optimal $(x_1, x_2)$ satisfies:

If on boundary $x_1 + x_2 = 1$, then $\lambda_1 > 0$, $\lambda_2 = 0$.

4. Applications

• Profit Maximization:

  • Use Lagrange multipliers for constrained optimization.
  • Example: Maximize $f(x,y) = 48x + 96y - x^2 - 2xy -9y^2 +1$ s.t. $20x +4y = 216$. Solve $\nabla f = \lambda \nabla g$ to find $x = 6$, $y = 24$, profit $= 541$.

2. Linear Algebra

3. Convexity & Optimization

4. Applications

Key Equations Summary