IIT M DIPLOMA AN EXAM QDD3 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. Exponential Distribution

   • PDF: $f_X(x) = \lambda e^{-\lambda x}$ (for $x \geq 0$)
   • CDF: $F_X(x) = 1 - e^{-\lambda x}$
   • Key Property: Memoryless ($P(X > s + t) = P(X > t)$).
   • Example: For $\lambda = 1$, $F_X(5) = 1 - e^{-5}$.

1. Gaussian (Normal) Distribution

   • PDF: $\mathcal{N}(\mu, \sigma^2)$
   • Linear Transformation: If $y \sim \mathcal{N}(\mu, \sigma^2)$ and $z = Cy + \epsilon$ where $\epsilon \sim \mathcal{N}(0, \sigma^2 I_d)$, then $p(z \mid y) = \mathcal{N}(Cy, \sigma^2 I_d)$.

1. Poisson Distribution

   • PMF: $P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$
   • Expectation: $E[X] = \lambda$.
   • Linearity of Expectation: For independent $X_1 \sim N(2,1)$ and $X_2 \sim \text{Poisson}(3)$, $E[3X_1 + 2X_2] = 3E[X_1] + 2E[X_2] = 3(2) + 2(3) = 12$.

1. Markov’s Inequality

   • For non-negative $Y$ with $E[Y] = 20$, $P(Y \geq 100) \leq \frac{E[Y]}{100} = 0.2$.

B. Linear Algebra

1. Singular Value Decomposition (SVD)

   • Definition: For any matrix $A \in \mathbb{R}^{m \times n}$, $A = U \Sigma V^T$, where:

     • $U$: Left singular vectors (orthogonal).
     • $\Sigma$: Diagonal matrix of singular values ($\sigma_i \geq 0$).
     • $V$: Right singular vectors (orthogonal).

   • Properties:

     • Singular values are non-negative (always true).
     • $A$ and $A^T$ share the same singular values.
     • SVD exists for any matrix (not just square/symmetric).

1. Principal Component Analysis (PCA)

   • Goal: Find orthogonal directions (principal components) that maximize variance.

   • Steps:

     1. Compute the covariance matrix $\Sigma$ of the data.

     2. Perform eigendecomposition of $\Sigma$: $\Sigma = V \Lambda V^T$, where:

        • Columns of $V$: Principal components (eigenvectors).
        • $\Lambda$: Eigenvalues (variances along PCs).
     3. Project data onto top-$k$ eigenvectors.

   • Key Points:

     • PCs are eigenvectors of the covariance matrix.
     • PCs form an orthogonal basis.
     • Maximizes variance sequentially.

1. Convex Sets & Functions

   • Convex Set: A set $S$ where for any $x, y \in S$ and $\theta \in [0,1]$, $\theta x + (1-\theta)y \in S$.

     • Examples:

       • Hyperplanes ($\sum a_i x_i = b$) are convex.
       • Lines in $\mathbb{R}^2$ are convex.
       • Non-examples: Unions of convex sets, arbitrary curves.

   • Convex Function: A function $f$ where for any $x, y$ and $\theta \in [0,1]$, $f(\theta x + (1-\theta)y) \leq \theta f(x) + (1-\theta)f(y)$.

     • Examples:

       • $e^x$, $e^{x^2+2}$, $x^2$.
       • Non-examples: $\cos(x)$, compositions of convex functions (not always convex).

C. Optimization

1. Gradient Descent

   • Update Rule: $x_{k+1} = x_k - \eta \nabla f(x_k)$, where $\eta$ is the learning rate.
   • Example: For $f(x,y) = x^2 + 2y^2 -6x +8y$, $\nabla f = (2x -6, 4y +8)$. At $(x_0, y_0) = (1, -2)$, $\nabla f = (-4, -8)$. With $\eta = 0.2$: $x_1 = 1 - 0.2(-4) = 1.8$, $y_1 = -2 - 0.2(-8) = -0.4$. Thus, $x_1 + y_1 = 1.4$. (Note: The answer in Q12 seems incorrect; the correct sum is 1.4.)

1. Lagrange Multipliers & KKT Conditions

   • General Form: For $\min f(x)$ subject to $g(x) = 0$ and $h(x) \leq 0$, the Lagrangian is: $L(x, \alpha, \beta) = f(x) + \alpha g(x) + \beta h(x)$.

   • KKT Conditions:

     1. Stationarity: $\nabla L = 0$.
     2. Primal Feasibility: $g(x) = 0$, $h(x) \leq 0$.
     3. Dual Feasibility: $\beta \geq 0$.
     4. Complementary Slackness: $\beta h(x) = 0$.

   • Example (Q13–Q15): For $\min x^2 + y^2 + z^2 + w^2 + 1$ with $x + y + z + w = 1$ and $w \leq \delta$, the KKT conditions imply:

     • $x = y = \frac{\alpha}{2}$, $z = w = \frac{\alpha}{2}$.
     • If $\delta > \frac{1}{4}$, the constraint $w \leq \delta$ is inactive ($\beta = 0$), so $x = y = z = w = \frac{1}{4}$.

D. Multivariable Calculus

1. Optimization with Constraints

   • Method: Use Lagrange multipliers or parameterization.

   • Example (Q11): Maximize $f(x,y,z) = 5x + 7y + z$ on the intersection of:

     • Plane: $x + y - z = 4$.
     • Cylinder: $x^2 + y^2 = 25$ (parameterized as $x = 5\cos t$, $y = 5\sin t$).
     • Substitute into $f$ and find critical points via $f'(t) = 0$.
     • Maximum value: 46.

2. Key Equations

Graph 2: Linear Algebra

Graph 3: Optimization

Graph 4: Convexity

4. Exam Tips

1. Probability: Focus on CDF/PDF properties, linear transformations of Gaussians, and expectation rules.
2. Linear Algebra: SVD/PCA properties are high-yield (e.g., singular values, eigenvectors).
3. Optimization: Know KKT conditions cold (stationarity, complementary slackness).
4. Convexity: Test functions using the definition (e.g., $e^x$ is convex; $\cos(x)$ is not).
5. Calculus: Parameterization (e.g., cylinders) and gradient descent updates are key.