Home / Past papers / MLF End Term / this paper

2023 Dec24: IIT M FN EXAM FDD2 - 24 Dec 2023

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

Elapsed 00:00

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:

Q5SA3 marks

Your numeric answer:

Q6SA3 marks

Your numeric answer:

Q7MSQ3 marksselect all that apply

Which of the following functions is/are continuous?

Q8MSQ3 marksselect all that apply

Let A be a n × n positive definite matrix. Then, which among the following are correct?

Q9SA4 marks

Your numeric answer:

Q10MSQ3 marksselect all that apply

Q11MCQ2 marks

Q12MCQ2 marks

Q13MCQ2 marks

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? Enter the answer correct to two decimal places.

Your numeric answer:

Q16SA2 marks

Your numeric answer:

Answer key

Q1 (MCQ, 0 marks): A
Q2 (SA, 3 marks): 9
Q3 (SA, 3 marks): 0.089 to 0.095
Q4 (SA, 3 marks): 0.74 to 0.79
Q5 (SA, 3 marks): 0.24 to 0.28
Q6 (SA, 3 marks): 0.020 to 0.030
Q7 (MSQ, 3 marks): A, B, D
Q8 (MSQ, 3 marks): A, B, D
Q9 (SA, 4 marks): 0.370 to 0.380
Q10 (MSQ, 3 marks): B, C
Q11 (MCQ, 2 marks): C
Q12 (MCQ, 2 marks): B
Q13 (MCQ, 2 marks): B
Q14 (MCQ, 2 marks): C
Q15 (SA, 2 marks): 5.30 to 5.37
Q16 (SA, 2 marks): 10

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.

Core Topics & Concepts

1. Probability & Random Variables

Discrete Distributions

Distribution	PMF ( $f_X(k)$ )	CDF ( $F_X(x)$ )	$E[X]$	$Var(X)$
Uniform( $A$ )	$\frac{1}{n}, \quad x=k$	$\frac{k-a+1}{n}$	$\frac{a+b}{2}$	$\frac{n^2-1}{12}$
Bernoulli( $p$ )	$p^x(1-p)^{1-x}$	$1-p$ (for $x<1$ )	$p$	$p(1-p)$
Binomial( $n,p$ )	${}^nC_k p^k (1-p)^{n-k}$	Cumulative sum of PMF	$np$	$np(1-p)$
Geometric( $p$ )	$(1-p)^{k-1}p$	$1-(1-p)^k$	$\frac{1}{p}$	$\frac{1-p}{p^2}$
Poisson( $\lambda$ )	$\frac{e^{-\lambda}\lambda^k}{k!}$	Cumulative sum of PMF	$\lambda$	$\lambda$

Continuous Distributions

Distribution	PDF ( $f_X(x)$ )	CDF ( $F_X(x)$ )	$E[X]$	$Var(X)$
Uniform( $[a,b]$ )	$\frac{1}{b-a}$	$\frac{x-a}{b-a}$	$\frac{a+b}{2}$	$\frac{(b-a)^2}{12}$
Exponential( $\lambda$ )	$\lambda e^{-\lambda x}$	$1-e^{-\lambda x}$	$\frac{1}{\lambda}$	$\frac{1}{\lambda^2}$
Normal( $\mu,\sigma^2$ )	$\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$	No closed form	$\mu$	$\sigma^2$
Gamma( $\alpha,\beta$ )	$\frac{\beta^\alpha x^{\alpha-1} e^{-\beta x}}{\Gamma(\alpha)}$	Incomplete Gamma	$\frac{\alpha}{\beta}$	$\frac{\alpha}{\beta^2}$
Beta( $\alpha,\beta$ )	$\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}$	Incomplete Beta	$\frac{\alpha}{\alpha+\beta}$	$\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$

Key Inequalities & Theorems

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(|\bar{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)

where $Y = \sum_{i=1}^n X_i$ and $X_i \sim \text{i.i.d.}$ .

Maximum Likelihood Estimation (MLE):

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

For Exponential( $\lambda$ ):

\hat{\lambda} = \frac{1}{\bar{X}}

2. Linear Algebra

Key Concepts

Positive Definite Matrices:
- $A$ is positive definite if $x^T A x > 0$ for all $x \neq 0$ .
- Properties:
  - All eigenvalues $> 0$ .
  - $A^k$ is positive definite for $k \geq 1$ .
  - $A^{-1}$ is positive definite.
  - Diagonal entries $> 0$ (but not sufficient alone).
Nullspace:
- For matrix $A$ , nullspace = $\{x | Ax = 0\}$ .
- Example: For $A = \begin{pmatrix}1&3&4\\2&3&5\\0&1&1\end{pmatrix}$ , nullspace is $\text{span}\{-1, -1, 1\}$ .
Eigenvalues & Diagonalizability:
- $c$ is an eigenvalue of $A$ iff $(A - cI)x = 0$ has a nontrivial solution.
- If $A$ has $n$ linearly independent eigenvectors, it is diagonalizable.

Optimization (Linear Programming)

Standard Form:

\text{Maximize } Z = c^T y \quad \text{subject to } Ay \leq b, y \geq 0

Example:

\text{Maximize } Z = 60y_1 + 45y_2 \quad \text{subject to } 2y_1 + 2y_2 \leq 200, y_1 + 3y_2 \leq 140

3. Machine Learning Basics

Linear Regression

Model: $f(\tilde{x}) = w_1 x_1 + w_2 x_2 + b$ .
Loss Function (MSE):

L = \frac{1}{n} \sum_{i=1}^n (f(\tilde{x}^i) - y^i)^2

(Note: Question 3 uses $-L$ , but standard MSE is minimized.)

Principal Component Analysis (PCA)

Steps:
1. Compute covariance matrix $C = \frac{1}{n} X^T X$ .
2. Find eigenvalues and eigenvectors of $C$ .
3. Project data onto top eigenvectors.
Projected Variance:
- For 1D PCA, variance = largest eigenvalue of $C$ .
- Example: For $X_1 = [-1,-1], X_2 = [1,1], X_3 = [3,3]$ , projected variance $\approx 5.33$ .

4. Convexity

Convex Functions:
- $f$ is convex if $f(\lambda x + (1-\lambda)y) \leq \lambda f(x) + (1-\lambda)f(y)$ .
- Examples: $f(x) = \frac{1}{x}$ (convex for $x > 0$ ), $g(x) = 1 + |x|$ (convex).
Composition Rules:
- If $f$ is convex and non-decreasing, and $g$ is convex, then $f \circ g$ is convex.
- Sum of convex functions is convex.

2023 Dec24: IIT M FN EXAM FDD2 - 24 Dec 2023

Answer key

Concepts this paper tests

Core Topics & Concepts

1. Probability & Random Variables

Discrete Distributions

Continuous Distributions

Key Inequalities & Theorems

2. Linear Algebra

Key Concepts

Optimization (Linear Programming)

3. Machine Learning Basics

Linear Regression

Principal Component Analysis (PCA)

4. Convexity

2. Linear Algebra

3. Machine Learning

4. Connections Between Topics