Home / Past papers / MLF End Term / this paper

IIT M FOUNDATION ET1 EXAM QPF2 S1 03 Sep - 03 Sep 2023

MLF End Term18 questions50 marks7 MSQ · 6 SA · 5 MCQFoundation / QPF2

Elapsed 00:00

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

Q18MCQ4 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
Q18 (MCQ, 4 marks): A

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.

1. Core Topics & Concepts

A. Probability & Statistics

1. Random Variables & Distributions

Discrete Distributions:
- Uniform (Discrete): PMF, CDF, Mean, Variance.
- Bernoulli: PMF, CDF, Mean (p), Variance (p(1-p)).
- Binomial: PMF (nCk p^k (1-p)^(n-k)), Mean (np), Variance (np(1-p)).
- Geometric: PMF ((1-p)^(k-1) p), Mean (1/p), Variance((1-p)/p^2).
- Poisson: PMF ((e^-λ λ^k)/k!), Mean (λ), Variance (λ).

Continuous Distributions:
- Uniform (Continuous): PDF (1/(b-a)), CDF, Mean ((a+b)/2), Variance ((b-a)^2/12).
- Exponential: PDF (λ e^(-λx)), CDF (1 - e^(-λx)), Mean (1/λ), Variance (1/λ^2).
- Normal: PDF ((1/(σ√(2π))) exp(-(x-μ)^2/(2σ^2))), Mean (μ), Variance (σ^2).
- Gamma: PDF ((β^α x^(α-1) e^(-βx))/Γ(α)), Mean (α/β), Variance (α/β^2).
- Beta: PDF (Γ(α+β)/(Γ(α)Γ(β)) x^(α-1) (1-x)^(β-1)), Mean (α/(α+β)).

2. Inequalities & Theorems

Markov’s Inequality: P(X ≥ c) ≤ μ/c (for non-negative X).
Chebyshev’s Inequality: P(|X - μ| ≥ kσ) ≤ 1/k^2.
Weak Law of Large Numbers (WLLN): P(|X̄ - μ| > δ) ≤ σ^2/(nδ^2) (convergence in probability).
Central Limit Theorem (CLT): (Y - nμ)/√(nσ) ~ N(0,1) (for i.i.d. samples Y = ΣX_i).

3. Maximum Likelihood Estimation (MLE)

Likelihood Function: L(θ) = Π f_X(x_i; θ).
MLE: θ* = argmax_θ L(θ).

4. Independence & Dependence

Pairwise Independence: P(A ∩ B) = P(A)P(B).
Mutual Independence: P(A ∩ B ∩ C) = P(A)P(B)P(C).
Example: Coin tosses (Q13) show pairwise independence but mutual dependence.

B. Linear Algebra

1. Matrix Properties

Orthogonal Matrices: A^T A = I (preserves norms, AB is orthogonal if A, B are).
Symmetric Matrices: A = A^T, eigenvalues real, eigenvectors orthogonal.
Eigenvalues/Eigenvectors:
- Av = λv.
- Symmetric matrices: A^T A and AA^T share non-zero eigenvalues.
Singular Values: Square roots of eigenvalues of A^T A.
Unitary Matrices: U^H U = I (complex analog of orthogonal).
Hermitian Matrices: A = A^H (self-adjoint).

2. Systems of Linear Equations

Consistency Condition: Ax = b has a solution iff b is in the column space of A.
- Example (Q6): 2b₁ - b₂ + b₃ = 0 for solvability.

3. PCA (Principal Component Analysis)

First PC: Direction of maximum variance (eigenvector of largest eigenvalue of covariance matrix).
Dimensionality Reduction: Project data onto top k eigenvectors.

C. Optimization

1. Gradient Descent

Update Rule: x_(k+1) = x_k - α ∇f(x_k).
Convergence:
- Convex Functions: Guaranteed to global optimum.
- Non-Convex: May converge to local minima.
Step Size (α): Too large → divergence; too small → slow convergence.
Example (Q14): 2 iterations of GD on f(x₁,x₂) = 8x₁² + 2x₁x₂ - x₂².

2. Constrained Optimization (KKT Conditions)

KKT Conditions (for min f(x) s.t. g_i(x) ≤ 0):
1. Stationarity: ∇f(x*) + Σ λ_i ∇g_i(x*) = 0.
2. Primal Feasibility: g_i(x*) ≤ 0.
3. Dual Feasibility: λ_i ≥ 0.
4. Complementary Slackness: λ_i g_i(x*) = 0.
Example (Q18): Minimize 3x + 2y with linear constraints.

3. Directional Derivatives & Gradients

Directional Derivative: D_v f = ∇f · v (rate of change in direction v).
Maximum Rate of Change: ||∇f|| (direction of gradient).
Example (Q4-Q5):
- Directional derivative of f(x,y,z) = x² + 2y² - 3z² at (1,1,1) in direction v.
- Maximum rate of change = √(∇f·∇f) ≈ 7.48.

D. Machine Learning Foundations

1. Classification vs. Regression

Aspect	Classification	Regression
Output	Categorical (discrete classes)	Continuous (real-valued)
Goal	Decision boundaries (e.g., SVM, logistic)	Predict numeric values (e.g., linear reg.)
Loss Functions	0-1 loss, cross-entropy	MSE, MAE

2. Probabilistic Models

Exponential Distribution: f(x) = λ e^(-λx) (memoryless property).
- Example (Q17): WLLN for bulb lifetimes (λ = 0.4).
Joint PDFs (Q19): Transformations of i.i.d. exponentials (Y₁ = X₁ + X₂, Y₂ = X₂).

2. Key Equations

Probability & Statistics

PMF/PDF, CDF, Mean, Variance for all distributions (see table in Q2).
Markov’s Inequality: P(X ≥ c) ≤ E[X]/c.
Chebyshev’s Inequality: P(|X - μ| ≥ kσ) ≤ 1/k².
WLLN: P(|X̄ - μ| > δ) ≤ σ²/(nδ²).
CLT: (Y - nμ)/√(nσ) ~ N(0,1).
MLE: θ* = argmax θ Π f(X_i; θ).

Linear Algebra

Orthogonal Matrix: A^T A = I.
Eigenvalue Equation: A v = λ v.
PCA: Covariance Matrix Σ = E[(X - μ)(X - μ)^T].
SVD: A = U Σ V^T.

Optimization

Gradient Descent: x_(k+1) = x_k - α ∇f(x_k).
Directional Derivative: D_v f = ∇f · v.
KKT Conditions:
- ∇f(x*) + Σ λ_i ∇g_i(x*) = 0,
- g_i(x*) ≤ 0, λ_i ≥ 0, λ_i g_i(x*) = 0.

IIT M FOUNDATION ET1 EXAM QPF2 S1 03 Sep - 03 Sep 2023

Answer key

Concepts this paper tests

1. Core Topics & Concepts

A. Probability & Statistics

1. Random Variables & Distributions

2. Inequalities & Theorems

3. Maximum Likelihood Estimation (MLE)

4. Independence & Dependence

B. Linear Algebra

1. Matrix Properties

2. Systems of Linear Equations

3. PCA (Principal Component Analysis)

C. Optimization

1. Gradient Descent

2. Constrained Optimization (KKT Conditions)

3. Directional Derivatives & Gradients

D. Machine Learning Foundations

1. Classification vs. Regression

2. Probabilistic Models

2. Key Equations

Probability & Statistics

Linear Algebra

Optimization

Machine Learning

Graph 2: Linear Algebra

Graph 3: Optimization

Graph 4: Machine Learning

Graph 5: Key Relationships