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IIT M FOUNDATION AN EXAM ADF3 24 Dec 2023 - 24 Dec 2023

MLF End Term16 questions40 marks8 MCQ · 5 SA · 3 MSQFoundation / afternoon

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Q1MCQ0 marks

Useful Data has been mentioned above.
This data attachment is just for a reference & not for an evaluation.

Q2SA3 marks

Your numeric answer:

Q3SA3 marks

Your numeric answer:

Q4SA3 marks

Your numeric answer:

Q5MCQ3 marks

Q6MCQ3 marks

Q7MCQ3 marks

Q8SA4 marks

Your numeric answer:

Q9MCQ2 marks

Q10MCQ2 marks

Which among the following statements are correct?

Q11MCQ2 marks

Q12MSQ3 marksselect all that apply

Q13MSQ3 marksselect all that apply

Comprehension passage - the questions below it refer to this

Q14MCQ2 marks

Which of the following represents the covariance matrix, C for the given data points?

Q15SA2 marks

What is the value of the projected variance?

Your numeric answer:

Q16MSQ2 marksselect all that apply

Answer key

Q1 (MCQ, 0 marks): A
Q2 (SA, 3 marks): 3.31 to 3.35
Q3 (SA, 3 marks): 0.5
Q4 (SA, 3 marks): 0.480 to 0.490
Q5 (MCQ, 3 marks): A
Q6 (MCQ, 3 marks): B
Q7 (MCQ, 3 marks): A
Q8 (SA, 4 marks): 1
Q9 (MCQ, 2 marks): B
Q10 (MCQ, 2 marks): B
Q11 (MCQ, 2 marks): D
Q12 (MSQ, 3 marks): A, B, D
Q13 (MSQ, 3 marks): B, C
Q14 (MCQ, 2 marks): C
Q15 (SA, 2 marks): 1
Q16 (MSQ, 2 marks): A, B

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 & Random Variables

Discrete Distributions

Distribution	PMF ( $f_X(k)$ )	$E[X]$	Var( $X$ )
Uniform( $a, b$ )	$\frac{1}{n}, \quad k=a,...,b$	$\frac{a+b}{2}$	$\frac{n^2-1}{12}$
Bernoulli( $p$ )	$p^k(1-p)^{1-k}, \quad k=0,1$	$p$	$p(1-p)$
Binomial( $n,p$ )	$^nC_k p^k (1-p)^{n-k}$	$np$	$np(1-p)$
Geometric( $p$ )	$(1-p)^{k-1}p$	$\frac{1}{p}$	$\frac{1-p}{p^2}$
Poisson( $\lambda$ )	$\frac{e^{-\lambda} \lambda^k}{k!}$	$\lambda$	$\lambda$

Continuous Distributions

Distribution	PDF ( $f_X(x)$ )	$E[X]$	Var( $X$ )
Uniform[ $a, b$ ]	$\frac{1}{b-a}, \quad a \leq x \leq b$	$\frac{a+b}{2}$	$\frac{(b-a)^2}{12}$
Exponential( $\lambda$ )	$\lambda e^{-\lambda x}$	$\frac{1}{\lambda}$	$\frac{1}{\lambda^2}$
Normal( $\mu, \sigma^2$ )	$\frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$	$\mu$	$\sigma^2$
Gamma( $\alpha, \beta$ )	$\frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}$	$\frac{\alpha}{\beta}$	$\frac{\alpha}{\beta^2}$

Key Theorems & Inequalities

Markov’s Inequality:

P(X \geq c) \leq \frac{E[X]}{c}, \quad X \geq 0

Chebyshev’s Inequality:

P(|X - \mu| \geq k\sigma) \leq \frac{1}{k^2}

Weak Law of Large Numbers (WLLN):

P(|\overline{X} - \mu| > \delta) \leq \frac{\sigma^2}{n\delta^2}

Central Limit Theorem (CLT):

\frac{Y - n\mu}{\sqrt{n}\sigma} \approx \text{Normal}(0, 1)

Maximum Likelihood Estimation (MLE):

\hat{\theta} = \arg \max_\theta \prod_{i=1}^n f_X(x_i; \theta)

2. Linear Algebra

Key Concepts

Nullspace: Solutions to $A\mathbf{x} = \mathbf{0}$ .
Eigenvalues & Eigenvectors:
- $A\mathbf{v} = \lambda \mathbf{v}$
- If $A$ is positive definite, $A^{-1}$ and $A^k$ are also positive definite.
Convex Hull: Smallest convex set containing all points.
- Any convex combination $\sum \alpha_i x_i$ (where $\sum \alpha_i = 1, \alpha_i \geq 0$ ) lies in the convex hull.

Key Equations

Linear Approximation (Multivariable):

f(x, y) \approx f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)

Covariance Matrix (for data points $X_1, ..., X_n$ ):

C = \frac{1}{n} \sum_{i=1}^n (X_i - \mu)(X_i - \mu)^T

PCA Projection:
- Project data onto the eigenvector of the largest eigenvalue of $C$ .

3. Optimization & Linear Programming

Key Concepts

Objective Function: Minimize/maximize a linear/nonlinear function.
Constraints: Inequalities/equalities restricting feasible solutions.
Example (Linear Programming):

\text{Minimize } 20x_1 + 30x_2 \quad \text{s.t.} \quad x_1 + x_2 \geq 250, \quad 6x_1 + 5x_2 \geq 1000

4. Dimensionality Reduction (Autoencoders & PCA)

Key Concepts

Encoder-Decoder Pair:
- Encoder: $f: \mathbb{R}^d \rightarrow \mathbb{R}^k$ (compression).
- Decoder: $g: \mathbb{R}^k \rightarrow \mathbb{R}^d$ (reconstruction).
- Reconstruction Error:

R(f, g) = \frac{1}{n} \sum_{i=1}^n \|X_i - g(f(X_i))\|^2

PCA:
- Find the direction (eigenvector) maximizing variance.
- Project data onto this direction.

5. Conditional Expectation & Joint Distributions

Key Equations

Conditional Expectation:

E[X|Y=y] = \int x f_{X|Y}(x|y) \, dx

Joint PDF/CDF:
- For independent $X, Y$ : $f_{X,Y}(x,y) = f_X(x)f_Y(y)$ .

2. Linear Algebra & Optimization

3. Dimensionality Reduction

4. Conditional Probability & Expectation

Key Equations Summary

Topic	Equation
Markov’s Inequality	$P(X \geq c) \leq \frac{E[X]}{c}$
Chebyshev’s Inequality	$P(\|X - \mu\| \geq k\sigma) \leq \frac{1}{k^2}$
WLLN	$P(\|\overline{X} - \mu\| > \delta) \leq \frac{\sigma^2}{n\delta^2}$
CLT	$\frac{Y - n\mu}{\sqrt{n}\sigma} \approx \text{Normal}(0, 1)$
MLE	$\hat{\theta} = \arg \max_\theta \prod_{i=1}^n f_X(x_i; \theta)$
Linear Approximation	$f(x, y) \approx f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)$
Reconstruction Error	$R(f, g) = \frac{1}{n} \sum_{i=1}^n \\|X_i - g(f(X_i))\\|^2$
PCA Variance	Maximize $\text{Var}(X^Tw) = w^T C w$ s.t. $\\|w\\| = 1$
Conditional Expectation	$E[X\|Y=y] = \int x f_{X\|Y}(x\|y) \, dx$

Exam Tips

Probability:
- Memorize PMF/PDF, expectation, and variance for key distributions.
- Practice applying Markov/Chebyshev inequalities and CLT.
Linear Algebra:
- Understand eigenvalues, nullspace, and convex hulls.
- Know how to compute covariance matrices and PCA projections.
Optimization:
- Formulate LP problems correctly (objective + constraints).
- Recognize when to use MLE (e.g., estimating $p$ for Bernoulli).
Dimensionality Reduction:
- Compute reconstruction error for autoencoders.
- Derive PCA steps (covariance matrix → eigenvectors → projection).
Conditional Probability:
- Use joint PDFs to compute conditional expectations.
- Remember independence implies $f_{X,Y}(x,y) = f_X(x)f_Y(y)$ .