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2023 Dec24: IIT M AN EXAM ADD3 - 24 Dec 2023

MLF End Term16 questions40 marks8 MCQ · 5 SA · 3 MSQafternoon

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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( $a, b$ )	$\frac{1}{n}, \quad n=b-a+1$	$\frac{a+b}{2}$	$\frac{n^2-1}{12}$
Bernoulli( $p$ )	$p^x(1-p)^{1-x}$	$p$	$p(1-p)$
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[ $a, b$ ]	$\frac{1}{b-a}$	$\frac{a+b}{2}$	$\frac{(b-a)^2}{12}$
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( $\alpha, \beta$ )	$\frac{\beta^\alpha x^{\alpha-1} e^{-\beta x}}{\Gamma(\alpha)}$	$\frac{\alpha}{\beta}$	$\frac{\alpha}{\beta^2}$

Key Theorems & Inequalities

Markov’s Inequality:

P(X \geq c) \leq \frac{E[X]}{c}, \quad c > 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, all eigenvalues are positive.
- $A^{-1}$ and $A^k$ share eigenvectors with $A$ .
Convex Hull: Smallest convex set containing all points in $S$ .
PCA (Principal Component Analysis):
- Projects data onto directions of maximum variance.
- Covariance matrix $C = \frac{1}{n-1} X^T X$ .

Optimization

Linear Programming:
- Minimize/maximize a linear objective subject to linear constraints.
- Example:

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

3. Multivariable Calculus

Linear Approximation

For $f(x, y)$ near $(a, b)$ :

L(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)

Example for $f(x, y) = e^{xy}$ at $(1, 1)$ :

L(x, y) = e + e(x - 1) + e(y - 1) = e(x + y - 1)

Conditional Expectation

For joint density $f(x, y)$ , the conditional expectation is:

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

4. Machine Learning Foundations

Dimensionality Reduction

Encoder-Decoder Pair:
- Encoder: $f(\mathbf{x}) = \frac{1}{3}(x_1 + x_2 + x_3)$ .
- Decoder: $g(u) = [u, u^2, u]$ .
- Reconstruction error:

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

PCA:
- Transform data to maximize variance along principal components.
- Projected variance is the largest eigenvalue of the covariance matrix.

Probabilistic Models

Joint & Conditional Probability:
- For independent $X \sim \text{Bernoulli}(p_1)$ , $Y \sim \text{Bernoulli}(p_2)$ :

P(X + Y = 1) = P(X=1)P(Y=0) + P(X=0)P(Y=1)

MLE for Bernoulli:
- Given $k$ heads in $n$ trials, $\hat{p} = \frac{k}{n}$ .

2. Linear Algebra & Optimization

3. Multivariable Calculus & ML

Key Equations to Remember

Reconstruction Error:

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

Linear Approximation:

L(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)

MLE for Bernoulli:

\hat{p} = \frac{\text{Number of successes}}{n}

PCA Projected Variance:

\text{Var}_{\text{projected}} = \lambda_1 \quad (\text{largest eigenvalue of } C)

Conditional Expectation:

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