IIT M DIPLOMA AN EXAM QDD3 31 Aug - 31 Aug 2025

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 the core topics, concepts, principles, and equations essential for this exam, followed by Mermaid knowledge graphs to visualize relationships.

Core Topics & Key Concepts

1. Probability & Statistics

• Markov’s Inequality: For a non-negative random variable $X$, $\mathbb{P}(X \geq a) \leq \frac{\mathbb{E}[X]}{a}$.

  • Applied to $X \sim \text{Binomial}(n,p)$: $\mathbb{P}(X > 2pn) \leq \frac{1}{2}$.

• Chebyshev’s Inequality: For any random variable $X$ with mean $\mu$ and variance $\sigma^2$, $\mathbb{P}(|X - \mu| \geq k\sigma) \leq \frac{1}{k^2}$.

  • For $X \sim \text{Binomial}(n,p)$: $\mathbb{E}[X] = np$, $\text{Var}(X) = np(1-p)$. $\mathbb{P}(|X - np| \geq pn) \leq \frac{1-p}{np}$.

• Joint Probability Density Function (PDF):

  • Compute marginal/conditional distributions, expectations, and variances.
  • Example: $f(x,y) = 24xy$ for $0 < x,y < 1$, $x+y < 1$.
  • Variance of $Y$: $\text{Var}(Y) = \mathbb{E}[Y^2] - (\mathbb{E}[Y])^2$.

• Normal Distribution:

  • Linear combinations of independent normals: If $X \sim N(\mu_X, \sigma_X^2)$, $Y \sim N(\mu_Y, \sigma_Y^2)$, then: $U = aX + bY \sim N(a\mu_X + b\mu_Y, a^2\sigma_X^2 + b^2\sigma_Y^2)$.
  • Standardize to $Z$-score: $P(U < c) = F_Z\left(\frac{c - \mu_U}{\sigma_U}\right)$.

• Conditional Probability:

  • $P(C \mid D) = \frac{P(C \cap D)}{P(D)}$.
  • Example: Permutations of 4 friends with constraints.

2. Linear Algebra

• Vector Spaces & Span:

  • $S = \text{span}(v_1)$: All linear combinations of $v_1$.
  • Projection of $v_2$ onto $S$: $\text{proj}_{v_1} v_2 = \left( \frac{v_2 \cdot v_1}{v_1 \cdot v_1} \right) v_1$.
  • Orthogonality: $v_1 \cdot v_3 = 0$.

• Eigenvalues & Matrices:

  • Hermitian Matrix: $A = A^H$ (conjugate transpose).
  • Unitary Matrix: $A^H A = I$.
  • Diagonalizable: $A$ has $n$ linearly independent eigenvectors.
  • Eigenvalues of $2A^2 - 3I$: If $\lambda$ is an eigenvalue of $A$, then $2\lambda^2 - 3$ is an eigenvalue of $2A^2 - 3I$. Product of eigenvalues = determinant = $\prod (2\lambda_i^2 - 3)$.

3. Optimization & Convexity

• Convex Functions:

  • A function $f$ is convex if its Hessian $\nabla^2 f \succeq 0$.

  • Examples:

    1. $f_1(x,y) = x^4 + x^2 - 2xy + y^2$ (Hessian is PSD).
    2. $f_3(x) = (x^2 - 3x + 2)(x^2 - 7x + 12)$ is convex on $(2,3)$.

• Constrained Optimization:

  • Primal Problem: Minimize $f(x,y) = x^2 + y^2$ (convex) subject to linear constraints.
  • Lagrangian: $L(x,y,u_1,u_2) = x^2 + y^2 + u_1(-x - 3y + 9) + u_2(-2x - y + 6)$.
  • KKT Conditions: Stationary: $\nabla L = 0$: $2x - u_1 - 2u_2 = 0$, $2y - 3u_1 - u_2 = 0$.

• Convex Sets:

  • A set $S$ is convex if $\forall x,y \in S$, $\theta x + (1-\theta)y \in S$ for $\theta \in [0,1]$.
  • Linear constraints (e.g., $-x -3y \leq -9$) define convex sets.

4. Applied Mathematics

• Quadratic Cost Minimization:

  • Minimize $C = 5x^2 + 2xy + 3y^2 + 500$ subject to $x + y = 39$.
  • Use substitution and calculus (set $\frac{dC}{dx} = 0$).

Key Equations

2. Linear Algebra

3. Optimization & Convexity

4. Applied Math (Cost Minimization)

Exam Preparation Tips

1. Probability:

   • Memorize Markov/Chebyshev inequalities and their applications to binomial distributions.
   • Practice joint PDF calculations (marginals, expectations, variances).
   • Understand linear combinations of normals and standardization.

1. Linear Algebra:

   • Master projections, orthogonality, and span.
   • Review eigenvalues, Hermitian/unitary matrices, and diagonalizability.

1. Optimization:

   • Know how to check convexity (Hessian test).
   • Write Lagrangians and KKT conditions for constrained problems.
   • Recognize convex sets from linear inequalities.

1. Applied Math:

   • Solve constrained quadratic minimization using substitution and calculus.