AKTU Mathematical Foundation AI, ML and Data Science (KAI051) Question Paper 2023-24

Note: Attempt all Sections. If you require any missing data, then choose suitably.

Section A

Q1. Attempt all questions in brief · 2 × 10 = 20 marks

1. Explain the reading and interpretation of bar graphs.
2. Using Chebyshev’s inequality, calculate the percentage of observations that would fall outside 3 standard deviations of the mean. (ii)89% (iii) 90%
3. Discuss the need of sampling.
4. Explain the use of chi square test in hypothesis testing.
5. Briefly explain Gibbs sampling. 3
6. Interpret the need of random number generator.
7. Discuss vector space.
8. Explain linear independence.
9. Differentiate between symmetric matrix and anti symmetric matrix.
10. Discuss eigen value and eigen vectors.

Section B

Q2. Attempt any three of the following · 10 × 3 = 30 marks

1. The following table shows the number of Maruti car sold by five dealers in a particular month:
   Dealer Saya Bagga DD Bhasin Competent
   Links motors Motors
Cars 60 40 20 15 10
   Sold
   Represent the above information by a bargraph.
2. Discuss Central limit theorem with applications.
3. Explain Metropolis Hastings algorithm.
4. Explain the proof of Cauchy-schwarz Inequality.
5. Explain orthogonal diagonalization with the help of an example.

Section C

Q3

Attempt any one part of the following · 10 × 1 = 10 marks

1. Three persons A,B,C have applied for a job in a private company. The chance of their selections is in the ration of 1:2:4. The probabilities that A,B and C can introduce changes to improve the profits of the company are 0.8, 0.5 and 0.3 respectively. If the change does not take place, find th1 probability that it is due to the appointment of C.
2. Determine the mean and variance of the random variable X having the following probability distribution.
   X=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

Attempt any one part of the following · 10 × 1 = 10 marks

1. Calculate the ANOVA coefficient for the following data:
   Plant Number Average Span S
Hibiscus 5 12 2
Marigold 5 16 1
Rose 5 20 4
2. Discuss the steps involved in dimensionality reduction using PCA.

Q5

Attempt any one part of the following · 10 × 1 = 10 marks

1. Explain the Joint distributions mentioned below: (i) Discrete Joint Distributions. (ii) Continuous Joint distributions. (iii) Multinomial Distribution.
2. The breakdown of ages of all visitors to a convention is given in the table below.
   Fabine wants to take a stratified sample of the visitors at the convention
   She choses a sample size of 80. Calculate how many people she will need to sample from each age group.
   Age Number of people
5-15 132
16-25 678
26-40 543
41-60 289
61+ 108

Q6

Attempt any one part of the following · 10 × 1 = 10 marks

1. Explain the Gram Schmidt Process.
2. Explain how to find a basis of vector space.

Q7

Attempt any one part of the following · 10 × 1 = 10 marks

1. Diagonalize the matrix -3 4 9
2. Show that the transformation T: V 2 ( R ) → V 2 ( R ) defined by T(a, b) = 5 (a + b, a) ∀ a, b ∈ R is a linear transformation.

Other years

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