AKTU Mathematical Foundation AI, ML and Data Science (BCAI051) Question Paper 2024-25

Note: Attempt all Sections. In case of any missing data; choose suitably.

Section A

Q1. Attempt all questions in brief · 2 × 7 = 14 marks

1. How is the mode defined in statistics?
2. Define skewness? How does it affect the distribution of data?
3. Describe sampling in the context of inferential statistics?
4. Define a pseudo-random number?
5. Describe a subspace of a vector space.
6. Define the kernel of a linear transformation.
7. With example explain the process of diagonalization of a matrix?

Section B

Q2. Attempt any three of the following · 7 × 3 = 21 marks

1. Explain the concept of dispersion and describe the various measures of dispersion such as range, variance, and standard deviation. How do these measures help in understanding the spread of data?
2. Describe estimation and discuss the difference between point estimation and interval estimation. How does estimation help in making decisions about population parameters based on sample data?
3. Discuss about the Gibbs sampling? How does it work in Markov Chain Monte Carlo?
4. Explain the Cauchy-Schwarz inequality in inner product spaces? Explain its significance and how it can be used to bound the inner product of two vectors. Provide an example illustrating the Cauchy-Schwarz inequality.
5. Define a linear transformation. Explain the conditions that a function T:V→W must satisfy to be considered a linear transformation, where V and W are vector spaces. Provide examples of linear transformations and non-linear transformations.

Section C

Q3

Attempt any one part of the following · 7 × 1 = 7 marks

1. Explain probability theory and its importance in statistical analysis. Define measures of probability such as probability mass functions and probability density functions.
2. Discuss Chebyshev’s inequality and explain how it can be used to estimate the probability that a random variable lies within a certain number of standard deviations from its mean.

Q4

Attempt any one part of the following · 7 × 1 = 7 marks

1. Discuss the difference between a t-test and a z-test. What are the assumptions underlying each test, and in what situations should each be used?
2. What is ANOVA (Analysis of Variance), and how does it help in comparing the means of three or more groups? Explain the concept of between-group variance and within-group variance?

Q5

Attempt any one part of the following · 7 × 1 = 7 marks

1. Discuss the inverse-transform method for generating random numbers. How does it work, and in what types of distributions is this method useful?
2. Describe Monte Carlo hypothesis testing. How is it used to evaluate statistical hypotheses, and what advantages does it offer over traditional hypothesis testing methods?

Q6

Attempt any one part of the following · 7 × 1 = 7 marks

1. Describe the Gram-Schmidt process. Explain how this process can be used to convert a set of linearly independent vectors into an orthonormal set.
2. Explain inner product in a vector space? Explain the properties of an inner product, such as linearity, symmetry, and positivity.

Q7

Attempt any one part of the following · 7 × 1 = 7 marks

1. Explain the concept of symmetric matrices. Discuss the properties of symmetric matrices, particularly their eigenvalues and eigenvectors.
2. Discuss the concept of change of basis in the context of linear transformations. Explain how a basis change affects the matrix representation of a linear transformation?

Other years

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