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IIT M FOUNDATION AN EXAM QDF3 23 Feb - 23 Feb 2025

MLT Quiz 114 questions40 marks7 SA · 5 MCQ · 2 MSQFoundation / afternoon / QDF3
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Q1MCQ2 marks
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  1. (1) > (2) > (3)
  2. (1) < (2) < (3)
  3. (3) < (1) < (2)
  4. (2) < (3) < (1)
Q2MCQ3 marks
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Q3MCQ3 marks
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Q4MSQ4 marksselect all that apply
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Q5MSQ3 marksselect all that apply
Which of the following statements is/are true?
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Q6SA3 marks
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Q7SA3 marks
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Q8SA3 marks
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Q9SA3 marks
Suppose you toss a coin four times and observe no heads. You then give the coin to your friend, who tosses it until the first head occurs. Your friend tosses the coin a total of five times. Let p denote the probability that the coin comes up heads. Find the maximum likelihood estimate of p.
Q10SA4 marks
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Comprehension passage - the questions below it refer to this
Based on the above data, answer the given subquestions. question figure
Q11MCQ2 marks
Among the vectors given below, choose a representative that has unit length.
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Q12SA3 marks
If standard PCA is performed on this dataset, what is the variance along the first principal component? Enter the answer correct to two decimal places.
Comprehension passage - the questions below it refer to this
Suppose that 100 items are sampled from a manufacturing process and 10 are found to be defective. Let p denote the proportion of defective items. Assume that the prior distribution for p is Beta(1, 1). Based on the above data, answer the given subquestions.
Q13MCQ2 marks
Select the correct options from the following:
  1. Posterior distribution for p is Beta(91, 11).
  2. Posterior distribution for p is Beta(11, 91).
  3. Posterior distribution for p is Bernoulli(0.1).
  4. Posterior distribution for p is Bernoulli(0.9).
Q14SA2 marks
Find the posterior mean. Enter the answer correct to two decimal places.

Answer key

  1. Q1 (MCQ, 2 marks): C
  2. Q2 (MCQ, 3 marks): C
  3. Q3 (MCQ, 3 marks): C
  4. Q4 (MSQ, 4 marks): A, D
  5. Q5 (MSQ, 3 marks): A, B, C
  6. Q6 (SA, 3 marks): 4.92 to 4.98
  7. Q7 (SA, 3 marks): 74 to 75
  8. Q8 (SA, 3 marks): 0.07 to 0.11
  9. Q9 (SA, 3 marks): 0.10 to 0.12
  10. Q10 (SA, 4 marks): 0.86 to 0.90
  11. Q11 (MCQ, 2 marks): B
  12. Q12 (SA, 3 marks): 43 to 44
  13. Q13 (MCQ, 2 marks): B
  14. Q14 (SA, 2 marks): 0.10 to 0.11

Concepts this paper tests

Auto-generated summary from the source site. Handy as a checklist; the lessons above explain each idea from scratch.

Summary of Core Topics, Concepts, Principles, and Equations

Core Topics and Concepts:

  1. K-means Clustering and Lloyd's Algorithm

    • Understanding the iterative process of K-means.
    • Sum of squared distances minimization.
    • Cluster assignment and mean updates.

  1. Principal Component Analysis (PCA)

    • Eigenvalues and eigenvectors.
    • Projection of data points onto principal components.
    • Variance along principal components.
    • Data compression and storage reduction.

  1. Kernel Methods and Valid Kernels

    • Kernel functions and their properties.
    • Examples of valid kernels.

  1. Matrix Properties

    • Trace of matrices.
    • Eigenvectors and eigenvalues of matrices XXTXX^T and XTXX^TX.

  1. Probability and Statistics

    • Maximum Likelihood Estimation (MLE).
    • Beta distribution and posterior mean.
    • Gaussian Mixture Models (GMM) and Expectation-Maximization (EM) algorithm.

  1. Data Representation and Storage

    • Data matrices and their shapes.
    • Storage space calculation and reduction.

Important Equations:

  1. Sum of Squared Distances in K-means:
∑i=1n∣∣xi−μzi∣∣2\sum_{i=1}^{n} ||x_i - \mu_{z_i}||^2
  1. Projection of Data Points onto Principal Components:
Projection of xi on the kth principal component=nλ\text{Projection of } x_i \text{ on the } k^{\text{th}} \text{ principal component} = \sqrt{n \lambda}
  1. Percentage Decrease in Storage Space:
Percentage Decrease=(Original Size−Compressed SizeOriginal Size)×100\text{Percentage Decrease} = \left( \frac{\text{Original Size} - \text{Compressed Size}}{\text{Original Size}} \right) \times 100
  1. Euclidean Distance:
Distance between x1 and x2=(x1−x2)T(x1−x2)\text{Distance between } x_1 \text{ and } x_2 = \sqrt{(x_1 - x_2)^T (x_1 - x_2)}
  1. Posterior Mean for Beta Distribution:
Posterior Mean=α+number of successesα+β+number of trials\text{Posterior Mean} = \frac{\alpha + \text{number of successes}}{\alpha + \beta + \text{number of trials}}
  1. Probability in Gaussian Mixture Models:
P(xi belongs to component k)=πkN(xi∣μk,σk2)∑j=1KπjN(xi∣μj,σj2)P(x_i \text{ belongs to component } k) = \frac{\pi_k \mathcal{N}(x_i | \mu_k, \sigma_k^2)}{\sum_{j=1}^{K} \pi_j \mathcal{N}(x_i | \mu_j, \sigma_j^2)}