(SEM V) THEORY EXAMINATION 2023-24 MATHEMATICAL FOUNDATION AI , ML AND DATA SCIENCE

Subject Code: KAI051

Subject Name: Mathematical Foundation of AI, ML, and Data Science

Course: B.Tech (Semester V)

Maximum Marks: 100

Time: 3 Hours

Exam Year: 2023–24

Sections: A, B, and C

SECTION A – Short Answer Questions (2 × 10 = 20 Marks)

Attempt all questions briefly.

Explain the reading and interpretation of bar graphs.

Using Chebyshev’s inequality, calculate the percentage of observations that would fall outside 3 standard deviations of the mean.

Options: (i) 11% (ii) 89% (iii) 90%

Discuss the need of sampling.

Explain the use of chi-square test in hypothesis testing.

Briefly explain Gibbs sampling.

Interpret the need of random number generators.

Discuss vector space.

Explain linear independence.

Differentiate between symmetric and anti-symmetric matrices.

Discuss eigenvalues and eigenvectors.
 

SECTION B – Medium-Length Questions (10 × 3 = 30 Marks)

Attempt any three of the following.

The table below shows the number of Maruti cars sold by five dealers — draw a bar graph:

Discuss the Central Limit Theorem and its applications.

Explain the Metropolis–Hastings algorithm.

Explain and prove the Cauchy–Schwarz Inequality.

Explain Orthogonal Diagonalization with an example.
 

SECTION C – Long/Analytical Questions (10 × 5 = 50 Marks)

Q3. Probability & Random Variables

a. Three persons A, B, C have selection chances in ratio 1:2:4. Their probabilities of improving profits are 0.8, 0.5, and 0.3. If no change occurs, find the probability that C was appointed.
OR
b. Determine the mean and variance of a random variable XXX with:
| X | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| P(X) | 0.15 | 0.10 | 0.10 | 0.01 | 0.08 | 0.01 | 0.05 | 0.02 | 0.28 | 0.20 |

Q4. ANOVA & PCA

a. Calculate the ANOVA coefficient for:

OR
b. Discuss the steps of dimensionality reduction using PCA.

Q5. Distributions & Sampling

a. Explain the following joint distributions:

Discrete Joint Distribution

Continuous Joint Distribution

Multinomial Distribution
OR
b. Given age distribution at a convention, select a stratified sample (n = 80) proportionally:

Q6. Vector Spaces

a. Explain the Gram–Schmidt Process for obtaining orthonormal bases.
OR
b. Explain how to find a basis of a vector space.

Q7. Linear Transformations & Diagonalization

a. Diagonalize the matrix

[300−349003]\begin{bmatrix} 3 & 0 & 0 \\ -3 & 4 & 9 \\ 0 & 0 & 3 \end{bmatrix}​3−30​040​093​​

OR
b. Show that T:V2(R)→V2(R)T: V_2(\mathbb{R}) \rightarrow V_2(\mathbb{R})T:V2​(R)→V2​(R) defined by T(a,b)=(a+b,a)T(a, b) = (a + b, a)T(a,b)=(a+b,a) is a linear transformation.
 

Key Topics to Prepare

Statistics & Probability: Sampling, Chebyshev’s inequality, Central Limit Theorem, ANOVA.

Linear Algebra: Vector spaces, Linear independence, Basis, Gram–Schmidt, Eigenvalues, Diagonalization.

AI/ML Foundations: Gibbs sampling, Metropolis–Hastings, Random numbers, PCA, Joint distributions.

Theoretical Proofs: Cauchy–Schwarz inequality, Orthogonal diagonalization.

Study Tips

Practice derivations (Cauchy–Schwarz, Orthogonal Diagonalization).

Revise probability laws — Bayes, Conditional, Expectation, Variance.

Understand sampling techniques — stratified, random, Gibbs.

Learn PCA process: standardization → covariance matrix → eigen decomposition → projection.

Focus on matrix operations — orthogonality, linear transformations, eigenvalues.