(SEM V) THEORY EXAMINATION 2024-25 MATHEMATICAL FOUNDATION AI, ML AND DATA SCIENCE

Course: B.Tech (Semester V)

Subject Code: BCAI051

Maximum Marks: 70

Duration: 3 Hours

Exam Year: 2024–25

Paper ID: 310915

Section A – Short Answer Questions (2 × 7 = 14 Marks)

Students must answer all 7 questions briefly:

Define mode in statistics.

Explain skewness and how it affects data distribution.

Describe sampling in inferential statistics.

Define a pseudo-random number.

Describe a subspace of a vector space.

Define the kernel of a linear transformation.

Explain the diagonalization of a matrix with an example. 

Section B – Descriptive Questions (Attempt any 3 × 7 = 21 Marks)

Explain dispersion and discuss measures such as range, variance, and standard deviation.

Discuss estimation, and the difference between point and interval estimation.

Explain Gibbs sampling and its working in Markov Chain Monte Carlo (MCMC).

Explain the Cauchy–Schwarz inequality in inner product spaces with an example.

Define a linear transformation and give examples of linear and non-linear transformations. 

Section C – Long Answer Questions (7 Marks each)

Q3 (Probability and Inequality)

(a) Explain probability theory, PMF, and PDF.

(b) Discuss Chebyshev’s inequality and its application in probability.

Q4 (Statistical Tests)

(a) Difference between t-test and z-test — assumptions and use cases.

(b) Explain ANOVA, its purpose, and the concepts of between-group and within-group variance.

Q5 (Random Numbers & Monte Carlo)

(a) Explain inverse-transform method for generating random numbers and its applications.

(b) Describe Monte Carlo hypothesis testing, its process, and advantages.

Q6 (Vector Space)

(a) Describe the Gram–Schmidt process to convert linearly independent vectors into an orthonormal set.

(b) Explain inner product and its properties — linearity, symmetry, and positivity.

Q7 (Matrix Theory)

(a) Discuss symmetric matrices, their properties, and eigenvalues/eigenvectors.

(b) Explain change of basis in linear transformations and its effect on matrix representation. 

Key Topics for Revision

Descriptive Statistics: Mode, Skewness, Dispersion, Variance, Standard Deviation

Probability Theory and Inequalities

Sampling & Estimation Methods

Random Numbers & Monte Carlo Methods

Linear Algebra: Subspace, Kernel, Linear Transformation, Gram–Schmidt, Inner Product, Diagonalization, Change of Basis

Statistical Testing: t-test, z-test, ANOVA