all_quizzes

Quiz 1

1

You have a fair coin that you toss eight times. What is the probability that you’ll get no more than seven heads? 

 

  • Probability is 1-chance of getting 8 heads
  • \(\frac{1}{2}^8 = \frac{1}{256}\)

2

You have a fair coin that you toss eight times. What is the probability that you’ll get exactly seven heads?

  • 7 different ways to get exactly seven heads
  • \(7 \cdot \frac{1}{256} = \frac{7}{256}\)
  • \(\frac{1}{2}^8 = \frac{1}{256}\) chance for each way

3

Let P(X) = 0.2, P(Y) = 0.4, P(X|Y) = 0.5. What is P(Y|X)? 

  • Bayes rule is \(P(A|B) = \frac{P(B|A)\cdot P(A)}{B}\)
  • \(P(Y|X) = \frac{0.5 \cdot 0.4}{0.2} = 1\)

4

Let P(X) = 0.2, P(Y) = 0.4. If P(X|Y) = 0.2, what can you say about X & Y?

  • Bayes rule can be used again, but this can also be reasoned out
  • \(P(X) = 0.2 = P(X|Y)\)
  • The probability of X is the same as the probability of X given Y
  • Y has no relationship to X
  • They are independent

5

Which of these numbers cannot be a probability?

0, 1.0, 1.5, 0.5

Probabilities are given as a chance of an event occurring against some other condition. This means that probabilities cannot be greater than 1 or less than 0.

  • 1.5

6

A die is rolled and a coin is tossed simultaneously. What is the probability of getting an even number on the die and a head on the coin?

  • Probability of even numbers on a 6 sided die is 1/2
  • probability of a head on a coin flip is 1/2
  • \(\frac{1}{2}^2 = \frac{1}{4}\)

7

Let \(f(x) = x^2\) What is its integral and differential?

  • Power rule of integration \(\int x^2dx = \frac{x^3}{3} + c\) Where c is an unknown constant.
  • Power rule of derivation \(\frac{df}{dx}x^2 = 2x\).

8

Let A be a 3x4 matrix of the following format

\(A = \begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 2 & 0 & 2 \\ 1 & 1 & 1 \end{bmatrix} \)

What is the rank of A?

  • Compute echelon form
  • \(A = \begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 2 & 0 & 2 \\ 1 & 1 & 1 \end{bmatrix} \rightarrow \begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \)
  • 2 non zero rows, rank=2

9

Quiz 2 explain

1. drill

A data-point in a dataset can be written as (1, 1, 1). What is the dimensionality of this dataset?

  • ☑ 3
  • ☐ 1
  • ☐ 2
  • ☐ 0

2. drill

Vectors are generally represented as:

  • ☑ column vector
  • ☐ linked-list
  • ☐ matrix
  • ☐ row vector

3. drill

Which of the following is true about a statistic?

  • ☑ Parameter of the population estimated from samples
  • ☐ Parameter of the population estimated by the entire population
  • ☐ geoemetric view in 1D
  • ☐ geometrix view in 2D

4. drill

Which of these statements is false

  • ☑ In geometric view, each attribute is a random variable
  • ☐ In geometric 2D, each data-point is a vector
  • ☐ In probablilitic view, parameters are estimated
  • ☐ For continuous attributes, mean of an attribute is expressed as an integration \(\int_{-\infty}^{\infty}xp\left(x\right)dx\)

5. drill

Which of the following is false

  • ☐ Correlation measures linear relationships
  • ☐ Cos(θ) is a measure of similarity
  • ☐ Euclidean distance is a good measure for geometric distances
  • ☑ Covariance is normalized correlation

Quiz 3 explain

1. drill

A matrix \(\Sigma\) is positive semidefinite if:

  • ☐ \( x^T\Sigma x\in\mathbb{Z}\)
  • ☐ \(x^T\Sigma x=0\)
  • ☑ \(x^T\Sigma x\geq0\)
  • ☐ \(x^T\Sigma x\leq 0\)

2. drill

The probability density of the Gaussian/Normal distribution is highest at

  • ☐ \(\mu + \sigma\)
  • ☐ \(\mu - \sigma\)
  • ☑ \(\mu\)
  • ☐ \(\mu + 2\sigma\)

3. drill

Similarity between pairs of categorical attributes can be obtained by

  • ☐ Correlation
  • ☐ Cosine
  • ☐ Covariance
  • ☑ \(\chi^{2}\) test

4. drill

CLT states that when random samples are drawn from any distribution:

  • ☐ The samples are uniformly distributed
  • ☑ The means of the samples are normally distributed
  • ☐ The means of the samples are uniformly distributed
  • ☐ The samples are normally distributed

5. drill

An attribute A takes 2 values {yes,no}, and attribute B takes 3 values {high,medium,low}. Which of the following is not true?

  • ☐ The confusion matrix is of size 2 x 3
  • ☑ If p-value of \(\chi^{2}\) test is 0.3 implies that A & B are independent
  • ☐ Since A & B are categorical, correlation is NOT the correct metric to measure similarity
  • ☐ The null hypothesis of \(\chi^2\) test is that variables are independent

Quiz 4 explain

1. drill

Your dataset has d binary attributes. Which of the following best describe the points?

  • ☐ The origin in d-dimensions
  • ☑ The corners of a d-dimensional hypercube
  • ☐ The surface of a d-dimensional hypershere
  • ☐ The shell of a d-dimensional hypersphere

2. drill

As \(d \rightarrow \infty,\) the volume of a unit hypershere goes to

  • ☐ \(\infty\)
  • ☐ 1
  • ☑ 0
  • ☐ e

3. drill

As \(d \rightarrow \infty\), which of the following is false?

  • ☑ The probability of sampling points near the origin is high
  • ☐ The volume of a unit hypercube is 1
  • ☐ The volume of a hypercube with sides of length 2 goes to ∞
  • ☐ The "corners" of a hypercube occupies more space than the inscribed hypercube

4. drill

In d-dimensional space, how many orthogonal axes do we have in addition to the major axes?

  • ☐ \(\mathcal{O}(d)\)
  • ☐ \(\mathcal{O}(d^2)\)
  • ☑ \(\mathcal{O}(2^d)\)
  • ☐ \(\mathcal{O}(d^3)\)

5. drill

A unit hypercube in 2D is best described as:

  • ☐ a line with length = 1
  • ☐ a circle with radius = 1
  • ☑ a square with side = 1
  • ☐ a circle with diameter = 1

Quiz 5 explain

1. drill

Let \(x_1,x_2,x_3 \) represent 3 features. Which of the following are NOT linear combinations of these features?

  • ☐ \(0.4x_1 + 0.3x_2 + 0.6x_3\)
  • ☑ \(4x_1^2 + 3x_2^2 + x_3^2\)
  • ☐ \(4^2 x_1 + 3^2 x_2 + 6^2 x_3\)
  • ☐ \(4x_1 + 3x_2 + 6x_3\)

Question 2 drill

Which one of the following statements about PCA is false?

  • ☐ PCA projects the attributes into a space where covariance matrix is diagonal
  • ☐ The first Principal Component points in the direction of maximum variance
  • ☑ PCA is a non-linear dimensionality reduction technique
  • ☐ PCA is useful for exploratory data analysis

Question 3 drill

Which one of the following statements about PCA is false?

  • ☑ PCA works well for circular data
  • ☐ The first PC points to maximum variance
  • ☐ PCA computes eigen-value eigen-vector decomposition of the covariance matrix
  • ☐ PCA works well for ellipsoidal data

Question 4 drill

The magnitude of vector x projected onto a unit vector u is

  • ☐ \(x \times u\)
  • ☐ \((x - \mu_x) \cdot (u - \mu_u)\)
  • ☑ \(x\cdot u\)
  • ☐ \(||x||||u||\)

Question 5 drill

Feature selection is:

  • ☑ selecting a subset of attributes
  • ☐ selecting principal components with maximum variance
  • ☐ combining many features into one
  • ☐ selecting principal components that are not orthogonal to each other

Quiz 6 explain

Covariance matrix, eigenvectors, eigenvalues

1.

If \(u_1, u_2, \dots, u_d\) , are eigenvectors (column vectors) of the covariance matrix \(\Sigma\), and \(\lambda_1, \lambda_2, \dots, \lambda_n\) are the eigenvalues, then:

  • ☐ \(\Sigma = \lambda_1 u_1^T u_1 + \lambda_2 u_2^T u_2 + \dots \lambda_d u_d^T u_d\)
  • ☐ \(\Sigma = \lambda_1 u_1^T + \lambda_2 u_2^T + \dots \lambda_d u_d^T\)
  • ☑ \(\Sigma = \lambda_1 u_1 u_1^T + \lambda_2 u_2 u_2^T + \dots \lambda_d u_d u_d^T\)
  • ☐ \(\Sigma = \lambda_1 u_1 + \lambda_2 u_2 + \dots \lambda_d u_d\)

Question 2 drill

The power method can determine (select the best answer)

  • ☑ All eigenvalues and eigenvectors by deflation
  • ☐ Eigen value/eigen vector corresponding to second-largest variance
  • ☐ Eigen value/eigen vector corresponding to largest variance
  • ☐ Eigen value/eigen vector corresponding to the smallest variance

Question 3 drill

If \(X^c \in \mathbb{R}^{n \times d}\) is a centered matrix and Σ its covariance matrix, which of the following is PCA?

  • ☑ \(\Sigma = V\Delta V^T\)
  • ☐ \(X = U\Delta V^T\)
  • ☐ \(\Sigma = U\Delta V^T\)
  • ☐ \(X = V\Delta V^T\)

question 4 drill

If \(X^c \in \mathbb{R}^{n \times d}\) is a centered matrix and Σ its covariance matrix, which of the following is SVD?

  • ☑ \(X = U\Delta V^T\)
  • ☐ \(\Sigma = U\Delta V^T\)
  • ☐ \(X = V\Delta V^T\)
  • ☐ \(\Sigma = V\Delta V^T\)

Question 5 drill

In Singular Value Decomposition, what does the matrix V represent?

  • ☑ Eigenvectors of covariance of attributes
  • ☐ Eigenvectors of covariance of data-points
  • ☐ Matrix of eigenvalues on diagonal
  • ☐ Deflated matrix after removing first Principal Component

Quiz 7 explain

LDA/PCA

1. drill

Which one of the following is not LDA?

  • ☐ \(\max \frac{|m_1 - m_2|}{s_1^2 + s_2^2}\)
  • ☐ \(\min \frac{s_1^2 + s_2^2}{(m_1 - m_2) . (m_1 - m_2)}\)
  • ☑ \(\min \frac{|m_1 - m_2|}{s_1^2 + s_2^2}\)
  • ☐ \(\max \frac{|m_2 - m_1|}{s_1^2 + s_2^2}\)

Question 2 drill

A dataset lies in d dimensions. Which one of the following is true (Choose best option)?

  • ☐ PCA and LDA project data to 1 < d' <= d dimensions
  • ☐ PCA projects data to 1 dimension, LDA projects data to 1 < d' <=d dimensions
  • ☑ PCA projects data to d' <= d and LDA projects data to 1 dimension
  • ☐ PCA and LDA project data to 1 dimension

Question 3 drill

Which of the following is true?

  • ☐ LDA inputs data only. PCA inputs data and labels
  • ☑ LDA inputs dataset and label. PCA inputs only dataset
  • ☐ Both PCA & LDA input dataset only
  • ☐ Both PCA & LDA input dataset and labels

Question 4 drill

A dataset lies in d dimensions. Which of the following is true of PCA & LDA?

  • ☐ Both methods project data to higher dimension
  • ☑ Both methods project data to lower dimension
  • ☐ Both maximize variance in ℝd
  • ☐ Both minimize variance in ℝd

Question 5 drill

Which of the following is a generalized eigenvector problem?

  • ☐ \(Ax = \lambda x\)
  • ☐ \(Ax = A^{-1}x\)
  • ☑ \(Ax = \lambda B x\)
  • ☐ \(Ax = x\)