IIT M DIPLOMA FN EXAM QDD1 13 Apr 2025 - 13 Apr 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 core topics, concepts, principles, and equations essential for the exam, followed by Mermaid knowledge graphs to visualize relationships.

1. Core Topics & Key Concepts

A. Probability & Statistics

1. Joint Probability Density Functions (PDFs)

• Transformation of Random Variables:

  • Given joint PDF $f_{X,Y}(x,y)$, find the joint PDF of transformed variables $U = g(X,Y)$, $V = h(X,Y)$.
  • Use the Jacobian determinant for transformations:

where $J$ is the Jacobian matrix of the inverse transformation.

• Example: For $U = X + Y$, $V = X - Y$, the Jacobian is:

• Support of $U, V$: Determine the region where $f_{U,V}(u,v)$ is non-zero (e.g., $|v| \leq u \leq 4 - |v|$).

2. Expected Value & Linearity

• For independent random variables $X_1, X_2, \dots, X_n$:

• Example: If $X_1 \sim \text{Uniform}(0,1)$, $X_2 \sim \text{Poisson}(3)$, $X_3 \sim \text{Exponential}(2)$, then:

3. Markov’s Inequality

• For a non-negative random variable $X$ with $E[X] = \mu$:

• Example: If $E[X] = 8$, then $P(X \geq 16) \leq 0.5$.

4. Cumulative Distribution Function (CDF)

• For a uniform distribution $X \sim \text{Uniform}(a,b)$:

• Example: For $X \sim \text{Uniform}(0,4)$, $F_X(2) = 0.5$.

B. Linear Algebra & Matrix Decompositions

1. Singular Value Decomposition (SVD)

• For a matrix $A \in \mathbb{R}^{m \times n}$:

where:

• $U$: Left singular vectors (eigenvectors of $AA^T$).
• $\Sigma$: Diagonal matrix of singular values (square roots of eigenvalues of $A^T A$).

• $V$: Right singular vectors (eigenvectors of $A^T A$).

  • Key Properties:
• Singular values $\sigma_i = \sqrt{\lambda_i(A^T A)}$.
• Rank of $A$ = Number of non-zero singular values.
• SVD is not unique for square matrices (depends on sign flips in $U$ and $V$).

2. Principal Component Analysis (PCA)

• Steps:

  1. Compute the covariance matrix $\Sigma$ of the data.
  2. Find eigenvectors $v_i$ and eigenvalues $\lambda_i$ of $\Sigma$.
  3. Project data onto the top $k$ eigenvectors (principal components) with the largest eigenvalues.

• Properties:

  • Principal components are orthogonal.
  • They capture maximum variance in the data.
  • The components are uncorrelated.

C. Optimization

1. Convexity

• Convex Functions:

  • A function $f$ is convex if for all $x, y$ and $\theta \in [0,1]$:

• Examples:

  • Convex: $e^{r+2}$, $x^2$, $x^4 - 4x^2 + 5$ (for $|x| \geq \sqrt{2}$).
  • Non-convex: $\sin x$, $\ln x$ (not convex everywhere).
  • Convex Sets:
• A set $S$ is convex if for all $x, y \in S$ and $\theta \in [0,1]$:

• Examples:

  • Convex: Lines, intersections of convex sets.
  • Non-convex: Arbitrary curves.

2. Gradient Descent

• Update Rule:

where $\eta$ is the learning rate.

• Example: For $f(x,y) = x^2 + 3y^2 -4x +6y$:

Starting at $(2, -1)$ with $\eta = 0.1$:

3. Karush-Kuhn-Tucker (KKT) Conditions

• For an optimization problem:

The KKT conditions are:

1. Stationarity: $\nabla f(x) + \sum \lambda_i \nabla g_i(x) + \sum \mu_j \nabla h_j(x) = 0$.
2. Primal Feasibility: $g_i(x) \leq 0$, $h_j(x) = 0$.
3. Dual Feasibility: $\lambda_i \geq 0$.

4. Complementary Slackness: $\lambda_i g_i(x) = 0$.

   • Example: For the problem:

The Lagrangian is:

KKT conditions include:

• $\nabla L = 0$ (stationarity).
• $\beta (w - \delta) = 0$ (complementary slackness).

D. Function Analysis

1. Critical Points & Extrema

• For $f(x) = x^4 - 4x^2 + 5$:

  • Find critical points by setting $f'(x) = 0$:

• Second Derivative Test:

• At $x = \sqrt{2}$: $f''(\sqrt{2}) = 16 > 0$ (local minimum).
• At $x = -\sqrt{2}$: $f''(-\sqrt{2}) = 16 > 0$ (local minimum).
• At $x = 0$: $f''(0) = -8 < 0$ (local maximum).

Graph 2: Linear Algebra & Matrix Decompositions

Graph 3: Optimization

Graph 4: Function Analysis

3. Key Equations Summary