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2023 Sep03: IIT M DEGREE ET1 EXAM QPE1 - 03 Sep 2023

MLF End Term17 questions46 marks7 MSQ · 6 SA · 4 MCQQPE1

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Q1MCQ0 marks

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

Q2MSQ2 marksselect all that apply

Which of the following statements accurately describes the key difference between classification and regression models?

In classification, the output variable is continuous and real-valued, while inregression, it is categorical.
Regression models are used for predicting probabilities, whereas classificationmodels focus on predicting absolute values.
Classification models aim to find decision boundaries to separate data intoclasses, while regression models seek to predict a numeric value.
In regression, the commonly used loss function is mean squared error, whilein classification, 0 -1 loss is typically employed.

Comprehension passage - the questions below it refer to this

Based on the above data, answer the given subquestions. question figure

Q3SA2 marks

Your numeric answer:

Q4SA1 mark

Your numeric answer:

Q5MCQ3 marks

b1 − b2 + b3 = 0
b1 + b2 + b3 = 0
2b1 − b2 + b3 = 0
2b1 + b2 + b3 = 0

Q6MCQ3 marks

A consumer wants to maximize his utility subject to some constraints. He consumes two goods g1 and g2 and the utility function is the sum of g1 and g2. His budget is Rs.100. The per unit price of goods g1 and g2 are Rs.3 and Rs.4 respectively. Chose the correct optimization problem.

Q7MSQ4 marksselect all that apply

A − I (where I is the identity matrix of the same size) is also orthogonal.
Both AB and BA are orthogonal.
An and Bn are orthogonal for every n > 1.
A and B are also symmetric matrices.

Q8MSQ4 marksselect all that apply

Which of the following statements is/are true for a real symmetric matrix A?

AT A and AAT have the same rank.
AT A and AAT have the same eigenvalues
AT A and AAT have the same eigenvectors.
AT A and AAT have the same singular values.

Q9MSQ3 marksselect all that apply

Which of the following matrices is/are Unitary but not Hermitian?

Q10MSQ3 marksselect all that apply

Which among the following is true for PCA?

The first principal component represents the direction along which thevariance of the dataset is maximized.
The first principal component represents the direction along which thevariance of the dataset is minimized.
The first principal component is the eigenvector corresponding to the smallesteigenvalue of the covariance matrix.
The first principal component is the eigenvector corresponding to the largesteigenvalue of the covariance matrix.

Q11MSQ3 marksselect all that apply

Which of the following statements regarding gradient descent is/are correct?

If the step size is chosen as 1, we may “not always” arrive as close to theoptimal solution even after many number of iterations.
Solution obtained for optimization problem is the local minimum of itsobjective function.
Solution obtained for optimization problem is the global minimum of itsobjective function.
Gradient descent converges to the global optimum in the case of convexfunctions.
Gradient descent does not converge to the global optimum in the case ofconvex functions.

Q12MSQ3 marksselect all that apply

Suppose a fair coin is tossed twice. Let A denote the event that the first toss is a tail. Let B denote the event that the second toss is a tail. Let C denote the event that out of the first two tosses, either both are heads or both are tails. Which among the following options are true? Select all that apply.

A, B and C are independent.
A, B and C are dependent.
A, B and C are pairwise independent.
P(A) = P(B) = P(C) = 1/2

Q13SA2 marks

Your numeric answer:

Q14SA3 marks

Your numeric answer:

Q15SA3 marks

Your numeric answer:

Q16SA3 marks

The average life (in years) of an electric bulb follows exponential distribution with parameter λ = 0.4. Using Weak Law of large numbers, find a lower bound on the probability that the mean life of a random sample of 50 such bulbs falls between 1.5 and 3.5 years. Enter your answer correct to three decimal places.

Your numeric answer:

Q17MCQ4 marks

Answer key

Q1 (MCQ, 0 marks): A
Q2 (MSQ, 2 marks): C, D
Q3 (SA, 2 marks): 0
Q4 (SA, 1 marks): 7.44 to 7.52
Q5 (MCQ, 3 marks): C
Q6 (MCQ, 3 marks): A
Q7 (MSQ, 4 marks): B, C
Q8 (MSQ, 4 marks): A, B, D
Q9 (MSQ, 3 marks): A, D
Q10 (MSQ, 3 marks): A, D
Q11 (MSQ, 3 marks): A, B, D
Q12 (MSQ, 3 marks): B, C, D
Q13 (SA, 2 marks): 225
Q14 (SA, 3 marks): 1
Q15 (SA, 3 marks): 40
Q16 (SA, 3 marks): 0.870 to 0.880
Q17 (MCQ, 4 marks): C

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 & Concepts

A. Machine Learning Foundations

1. Classification vs. Regression

Classification:
- Output: Categorical (discrete classes).
- Goal: Find decision boundaries to separate classes.
- Loss functions: 0-1 loss, cross-entropy, log loss.
Regression:
- Output: Continuous (real-valued).
- Goal: Predict a numeric value.
- Loss functions: Mean Squared Error (MSE), Mean Absolute Error (MAE).

2. Principal Component Analysis (PCA)

First Principal Component (PC1):
- Direction of maximum variance in data.
- Eigenvector of the covariance matrix with the largest eigenvalue.
Subsequent PCs:
- Orthogonal to previous PCs, capturing decreasing variance.

3. Gradient Descent (GD)

Update Rule:

\theta_{new} = \theta_{old} - \eta \nabla J(\theta)

where:

$\eta$ : Learning rate (step size).
$\nabla J(\theta)$ : Gradient of the loss function.
- Convergence:
Convex functions: GD converges to global optimum.
Non-convex functions: May converge to local minima.
- Challenges:
If $\eta$ is too large → divergence.
If $\eta$ is too small → slow convergence.

4. Optimization Problems

Linear Programming (LP):
- Minimize/maximize a linear objective subject to linear constraints.
Karush-Kuhn-Tucker (KKT) Conditions:
- Necessary for optimality in constrained optimization.
- Conditions:
  1. Stationarity: $\nabla f(x) + \sum \lambda_i \nabla g_i(x) = 0$ .
  2. Primal Feasibility: $g_i(x) \leq 0$ .
  3. Dual Feasibility: $\lambda_i \geq 0$ .
  4. Complementary Slackness: $\lambda_i g_i(x) = 0$ .

B. Probability & Statistics

1. Probability Distributions

Distribution	PMF/PDF	Expectation $E[X]$	Variance $\text{Var}(X)$
Uniform (Discrete)	$f_X(k) = \frac{1}{n}$ for $k = a, a+1, \dots, b$	$\frac{a+b}{2}$	$\frac{n^2-1}{12}$
Bernoulli	$f_X(k) = p^k (1-p)^{1-k}$ for $k \in \{0,1\}$	$p$	$p(1-p)$
Exponential	$f_X(x) = \lambda e^{-\lambda x}$	$\frac{1}{\lambda}$	$\frac{1}{\lambda^2}$

2. Independence of Events

Pairwise Independence: $P(A \cap B) = P(A)P(B)$ .
Mutual Independence: $P(A \cap B \cap C) = P(A)P(B)P(C)$ .
Example: Coin tosses (Q13) show pairwise independence but not mutual independence.

3. Weak Law of Large Numbers (WLLN)

Statement:

\lim_{n \to \infty} P\left( \left| \frac{1}{n} \sum_{i=1}^n X_i - \mu \right| \geq \epsilon \right) = 0

Application: Estimating sample mean convergence to true mean.

4. Multivariate Normal Distribution

Covariance Matrix ( $\Sigma$ ):

\Sigma = \begin{pmatrix} \sigma_1^2 & \rho \sigma_1 \sigma_2 \\ \rho \sigma_1 \sigma_2 & \sigma_2^2 \end{pmatrix}

Independence Condition:
- If $Y_1$ and $Y_2$ are linear combinations of $X_1, X_2$ , they are independent iff their covariance is zero.

C. Linear Algebra

1. Orthogonal Matrices

Properties:
- $A^T A = I$ (and $A A^T = I$ ).
- $\det(A) = \pm 1$ .
- Product of orthogonal matrices is orthogonal.
- Not necessarily symmetric (unless $A = A^T$ ).

2. Symmetric Matrices

Properties:
- $A = A^T$ .
- Eigenvalues: Real.
- Eigenvectors: Orthogonal.
- $A^T A$ and $A A^T$ have:
  - Same eigenvalues.
  - Same singular values.
  - Same rank.

3. Unitary Matrices

Definition: $U^H U = I$ (where $U^H$ is the conjugate transpose).
Hermitian Matrices: $A = A^H$ (not all unitary matrices are Hermitian).

4. Solving Linear Systems

Condition for Solution Existence:
- For $A \mathbf{x} = \mathbf{b}$ , a solution exists iff $\mathbf{b}$ is in the column space of $A$ .
- Example (Q6): $2b_1 - b_2 + b_3 = 0$ .

D. Calculus & Optimization

1. Directional Derivative

Formula:

D_{\mathbf{v}} f(\mathbf{p}) = \nabla f(\mathbf{p}) \cdot \mathbf{v}

where $\mathbf{v}$ is a unit vector.

Maximum Rate of Change:
- Occurs in the direction of $\nabla f$ .
- Magnitude: $\|\nabla f\|$ .

2. Gradient Descent Example (Q14)

Function: $f(x_1, x_2) = 8x_1^2 + 2x_1 x_2 - x_2^2$ .
Gradient:

\nabla f = \begin{pmatrix} 16x_1 + 2x_2 \\ 2x_1 - 2x_2 \end{pmatrix}

Update Rule:

\mathbf{x}_{new} = \mathbf{x}_{old} - \eta \nabla f(\mathbf{x}_{old})

3. Distance Minimization (Q15)

Problem: Minimize distance from origin to $x^2 - y^2 - z^2 = 1$ .
Solution: Use Lagrange multipliers or recognize it as a hyperboloid with minimum distance 1.

2. Key Equations

Probability & Statistics

Exponential Distribution (Q17):

P\left( \left| \frac{1}{n} \sum X_i - \mu \right| < \epsilon \right) \geq 1 - \frac{\sigma^2}{n \epsilon^2}

Covariance for Independence (Q16):

\text{Cov}(Y_1, Y_2) = 0 \implies \text{Independent if jointly normal}

Linear Algebra

Orthogonal Matrix Property (Q8):

A^T A = I

Symmetric Matrix Eigenvalues (Q9):

A = A^T \implies \text{Eigenvalues are real}

Calculus

Directional Derivative (Q4, Q5):

D_{\mathbf{v}} f = \nabla f \cdot \mathbf{v}

Gradient Descent Update (Q12, Q14):

\theta_{t+1} = \theta_t - \eta \nabla J(\theta_t)

Optimization

KKT Conditions (Q18):

\nabla f(x) + \sum \lambda_i \nabla g_i(x) = 0

Lagrange Multipliers (Q15):

\nabla f = \lambda \nabla g

2023 Sep03: IIT M DEGREE ET1 EXAM QPE1 - 03 Sep 2023

Answer key

Concepts this paper tests

1. Core Topics & Concepts

A. Machine Learning Foundations

1. Classification vs. Regression

2. Principal Component Analysis (PCA)

3. Gradient Descent (GD)

4. Optimization Problems

B. Probability & Statistics

1. Probability Distributions

2. Independence of Events

3. Weak Law of Large Numbers (WLLN)

4. Multivariate Normal Distribution

C. Linear Algebra

1. Orthogonal Matrices

2. Symmetric Matrices

3. Unitary Matrices

4. Solving Linear Systems

D. Calculus & Optimization

1. Directional Derivative

2. Gradient Descent Example (Q14)

3. Distance Minimization (Q15)

2. Key Equations

Probability & Statistics

Linear Algebra

Calculus

Optimization

Graph 2: Probability & Statistics

Graph 3: Linear Algebra

Graph 4: Calculus & Optimization

Graph 5: Relationships Between Topics