AKTU Mathematical Foundation AI, ML and Data Science (BCAI051) Question Paper 2025-26

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. Define the expectation 𝑬ሺ𝑿ሻ of a random variable and state one of its properties.
2. What is skewness? Why is it important in data analysis?
3. Define hypothesis testing.
4. What is resampling in statistics?
5. Define a vector space.
6. What is Principal Component Analysis (PCA)?
7. State the Markov property.

Section B

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

1. The following data represents the heights (cm) and weights (kg) of 10 individuals : Heights: 150 155 160 162 165 168 170 172 175 178 Weights: 52 55 58 60 62 64 66 68 70 72 (i) Compute the Pearson correlation coefficient ሺ𝒓ሻ using (ii) Test the significance of correlation at 𝜶 ൌ 𝟎. 𝟎𝟓 . (iii) Interpret the result for predictive modeling.
2. A discrete random variable 𝑿 has the following probability distribution: 𝑿 0 1 2 3 𝑷ሺ𝑿ሻ 0.1 0.3 0.4 0.2 (i) Verify that the given distribution is valid. (ii) Find the expected value 𝑬ሺ𝑿ሻ and variance 𝑽𝒂𝒓ሺ𝑿ሻ using 𝑬ሺ𝑿ሻ ൌ 𝜮 𝒙 ൉ 𝑷ሺ𝑿ሻ 𝒂𝒏𝒅 𝑽𝒂𝒓ሺ𝑿ሻ ൌ 𝑬ሺ𝑿²ሻ െ ሾ𝑬ሺ𝑿ሻሿ² . (iii) Verify Chebyshev’s inequality for 𝒌 ൌ 𝟐 and comment on its significance when the underlying distribution is unknown.
3. A random sample of 𝒏 ൌ 𝟏𝟐 observations have sample mean and standard deviation s = 4.5 . (i) Construct a 95% confidence interval for the population mean using . (ii) Explain how confidence level affects the width of the interval.
4. The following contingency table shows the classification outcome of a machine learning model:
5. Explain the principle of Inverse Transform Sampling and show how an exponential random variable can be generated using 𝑿 ൌ െሺ𝟏 / 𝝀ሻ 𝒍𝒏ሺ𝟏 െ 𝑼ሻ , where 𝑼 ~ 𝑼𝒏𝒊𝒇𝒐𝒓𝒎ሺ𝟎, 𝟏ሻ .

Section C

Q3

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

1. Using Monte Carlo simulation , estimate ଴ ሺଵା𝐱²ሻ (i) Describe the random sampling procedure. (ii) Construct the Monte Carlo estimator. (iii) Discuss variance and convergence behavior.
2. Using the Gram–Schmidt orthogonalization process, obtain an orthonormal basis for the vectors: (1, 1, 0), (1, 0, 1), (0, 1, 1) . (i) Show all intermediate steps. (ii) Explain the importance of orthonormal bases in numerical stability.

Q4

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

1. Explain the concept of Analysis of Variance (ANOVA). Discuss the underlying assumptions of ANOVA, the logic of partitioning total variation into between-group and within-group variation, and explain how ANOVA is used to test the significance of differences among multiple population means in data science and machine learning applications.
2. Explain Singular Value Decomposition (SVD) of a matrix defined as 𝑨 ൌ 𝑼𝜮𝑽ᵀ . (i) Explain the significance of matrices 𝑼 , 𝜮 , and 𝑽 . (ii) Analyze how SVD is used for dimensionality reduction and noise remova l in machine learning and data science applications.

Q5

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

1. Explain the Metropolis–Hastings algorithm used in Markov Chain Monte Carlo (MCMC) methods. (i) Describe the role of proposal distribution 𝒒ሺ𝒙′ | 𝒙ሻ . (ii) Explain the acceptance probability given by 𝜶 ൌ 𝒎𝒊𝒏ሺ𝟏, 𝝅ሺ𝒙′ሻ𝒒ሺ𝒙 | 𝒙′ሻ / ሾ𝝅ሺ𝒙ሻ𝒒ሺ𝒙′ | 𝒙ሻሿሻ . (iii) Discuss convergence issues and challenges in high-dimensional probability spaces.
2. Discuss how numerical summaries (mean, variance, correlation) combined with graphical summaries (histograms, boxplots, scatter plots) help in interpreting and validating results of AI and ML models.

Q6

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

1. Perform a one-sample t-test to test whether the population mean is μ = 50 , given the sample data: 48, 52, 55, 49, 51, 54, 50, 53, 52, 51. (i) State hypotheses. (ii) Compute the test statistic. (iii) Draw conclusion at 𝜶 ൌ 𝟎. 𝟎𝟓 .
2. Explain Type-I and Type-II errors with reference to hypothesis testing in machine learning model evaluation.

Q7

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

1. Explain change of basis in vector spaces and illustrate how it improves numerical stability in computations.
2. Discuss the importance of customized numerical and graphical summaries in presenting results of AI/ML analyses.

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