Which of the following matrices can be used to determine the singularity of the system of equations below?

2x+3y=15

2x+4y=16

Consider the next three plots below.

Now, consider the next three system of equations below.

System 1 :

3x−2y=1 x+y=3

​ System 2 :

3x+3y=2 x+9y=6

System 3

x+3y=4 x+3y=3

Each plot represents one of the systems described. Choose the correct option.

Question 1

Calculate the determinant of the following matrix. Is the matrix singular or non-singular?

A=[2 3

2 4]

Hint: To find the determinant apply the formula ad−bc. A matrix of determinant 0 is singular, while a determinant different than 0 represents a complete system, thus a non-singular matrix.

Determine if this matrix has linearly dependent or independent rows.

[1 2

2 3]

A company produces three items: aprons, bags and coasters. The company wants to know how long it takes to produce each item.

• On the first day, the company spent 5 hours to make 5 aprons, 10 bags, and 10 coasters.

• On the second day, the company spent 7 hours to make 10 aprons, 5 bags, and 15 coasters.

• On the third day, the company spent 6 hours to make 4 aprons, 6 bags, and 5 coasters.

Which of the following systems of equations represents the correct information in the above system of sentences?

Consider the following system of equations:

3x+2y+z = 10

x+y+2z = 5

5x−6y+3z = 2

Which of the following matrices can be used to study the singularity of the system of equations above?

Calculate the determinant of the following matrix:

1 2 1

2 1 1

−1 2 1

Determine if the following matrix has linearly dependent or independent rows.

1 2 3

3 2 1

2 2 1

Consider the following matrix.

2 1 5

1 2 1

x y z

For which values x, y and z does the matrix have linearly dependent rows?

Calculate the determinant of the following matrix.

A= 1 2 3

0 2 2

1 4 5

Select which of the following are true for non-singular matrices.

In a non-singular matrix, one row can be a multiple of another one.

In a non-singular matrix, rows are linearly independent.

Correct

Non-singular matrices have linearly independent rows.

In a non-singular matrix, rows are linearly dependent.

In a non-singular matrix there is only a unique solution for the represented system of equations.

Solve the following system of equations using the method of elimination and select the correct answer:

х+у=4

-6х+2у=16

Choose the sequence of lines that represents a linear system such that the systems have, in this order:

1. Zero solution

2. Just one solution

3. Infinitely many solutions

Calculate the determinant of the following matrix and determine if it is singular or non-singular:

-3 8 1

2 2 -1

-5 6 2

Calculate the determinant of the following matrix and determine if it is singular or non-singular:

A= 4 -3

7 -8

Determine if the provided matrix has linearly dependent or independent rows (a,b,c,d,e,f are real numbers):

a b c

d e f

2a-d 2b-e 2c-f

Hint: Can one row in the matrix be obtained as a result of operations on the other rows?

Select the correct sequence of graphs that represents a linear system with, respectively:

1. Zero solutions

2. Just one solution

3. Infinitely many solutions

Let T be a linear transformation in the plane represented by the following matrix:

[1 0

2 3]

The rank of T is:

Which of the following operations, when applied to the rows of the matrix, do not change the singularity (or non-singularity) of the matrix:

Let M and N be two square matrices with the same size.

Check all statements that are true.

In the following matrix:

x x

y z

x, y, and z are non-zero real numbers. If the matrix is non-singular, which of the following must be true:

Let M be a square matrix.

Check all that are true.

What is the span of the following vectors?

[2 1 1], 

[1 0 2], 

[1 0 1]

Which of the following options is true for a vector?

Consider the following matrix:

M=[3 2

5 8]

The covariance matrix related to this matrix is:

Select the characteristic polynomial for the given matrix.

M=[2 1

3 6]

Compute the sum of the vectors $$\overrightarrow{u}$$ and $$\overrightarrow{v}$$ Hint: The sum vector is the diagonal in a parallelogram formed by the two vectors, $$\overrightarrow{u}$$ =(1,3) and $$\overrightarrow{v}$$ =(6,2).

Suppose you have the following dataset

Size (m2)No. BedroomsNo. BathroomsHouse 17022House 211042House 1House 2​Size (m2)70110​No. Bedrooms24​No. Bathrooms22​​

Which matrix is the X−μX−μ matrix, used in the covariance matrix computation? The matrix X−μX−μ is defined in the lecture Covariance Matrix. Remember that the covariance matrix is defined by Σ=1n−1(X−μ)T(X−μ)Σ=n−11​(X−μ)T(X−μ).

Consider the following matrix

M=[3 0

−2 1]

Check all the options that represent the eigenvectors of this matrix.

Compute the difference of the vectors $$\overrightarrow{u}$$ and $$\overrightarrow{v}$$

Calculate the dot product of the given vectors $$\overrightarrow{a}\cdot\overrightarrow{b}$$ and select the correct answer.

$$\overrightarrow{a\ }$$ =[-1 $$\overrightarrow{b}$$ =[-3

5 6

2 ] -4]

Which matrix is the X−μ matrix, used in the covariance matrix computation?

For the dataset from question 9, what are the eigenvalues of the covariance matrix?

Select all the options that are a basis for the 3D

Which of the following options is true for a vector?

Which of the following is the correct representative system equation for the given dot product?

Which of the following is true, if $$\overrightarrow{a}\cdot\overrightarrow{a}$$ =0 and $$\overrightarrow{a}\cdot\overrightarrow{b}$$ =0?

Let y1=ax+by1​=ax+b and y2=cx+dy2​=cx+d, where a,b,c,d∈Ra,b,c,d∈R. Check all the sentences that are true.

Which of the following sentences are true (check all that apply)?

What can be said about their slopes at their intersection?

What can be said about the curve’s slopes at the red point, which we will call P1, P2, P3 and P4, corresponding to the red points in the graphs 1, 2, 3 and 4, respectively?

Which of the following represents the derivative of a function f(x)f(x) (check all that apply)?

Regarding its derivative, f′(x), where x∈[0,5]: (check all that apply)

What is the derivative of $$3x^3-2x+1$$ ?

Suppose you have a game where you toss a coin 2020 times and win if you get, in this exact order, 16 heads and 4 tails. However, in this game, you can choose any coin and toss it 20 times.

Which of the following functions you need to maximize in order to find the best coin for this game? Consider p being the probability of a given coin being heads.

If f(x) and g(x) are differentiable functions, then the derivative of f(x)g(x) is given by:

Using the chain rule, the derivative of $$e^{-x}$$ is:

Let f(x) be a positive real function and g(x)=log f(x)

Question 2

Given that f(x ,y)=x*y2+2x+3y its gradient, i.e., ∇f(x,y) is:

The gradient of f(x, y ,z)= $$x^2+2x\cdot y\cdot z+z^2$$

About the Gradient Descent method, choose all that are true:

Given the Initial Point on the following graph, to which point will the Gradient Descent method converge?

Let f(x ,y)=2x^2+3y^2−2*x*y−10x, the minimum value of f(x ,y) is

What are the parameters that the Gradient Descent algorithm has? (check all that apply)

Let f(x ,y)=x^2+y^2−6x and ∇f(x ,y)=[2x−6 2y​] and let the initial point x0=(0,1). Performing the gradient descent algorithm with learning rate = 0.1, the first iteration will lead us the point x1x1​ which is:

Using Newton’s method, find an approximation recursive formula for $$\sqrt[]{2}$$

Regarding the previous question, suppose you don’t know any approximation for $$\sqrt[]{2}$$ and only that it is a positive real number such that x^2=2. Which value from the list below will result in the fastest convergence?

Let’s continue investigating the method we are developing to compute the $$\sqrt[]{2}$$ . Remember that we used the fact that $$\sqrt[]{2}$$ ​ is one of the roots of $$x^2-2$$ . What would happen if we have chosen a negative value as initial point?

Did you know that it is possible to calculate the reciprocal of any numberwithout performing division? (The reciprocal of a non-zero real number a is $$\frac{1}{a}$$ ​).

Setting a non-zero real number aa, use the function $$f\left(x\right)=a\ -\ \frac{1}{x}=a-x^{-1}$$ to find such formula.

This method was in fact used in older IBM computers to implement division in hardware!

So, the iteration formula to find the reciprocal of a, in this case, is:

Suppose we want to find the minimum value (suppose we already know that the minimum exists and is unique) of x*log⁡(x) where x∈(0,+∞). Using Newton’s method, what recursion formula we must use?

Regarding the Second Derivative Test to decide whether a point with f’(x)=0f’(x)=0 is a local minimum or local maximum, check all that apply.

How many parameters has a Neural Network with:

• Input layer of size 3

• One hidden layer with 3 neurons

• One hidden layer with 2 neurons

• Output layer with size 1

An image is provided below:

Let f(x ,y)=x^2+y^3, then the Hessian matrix,H(x ,y) is:

Given the following Single Layer Perceptron with Sigmoid function as activation function, and log-loss as Loss Function (L), the value for $$\frac{\partial L}{\partial\omega_1}$$ ​ is:

Suppose you have a function f(x ,y) with ∇f(x0,y0)=(0,0) and such that

H(x0,y0)=[2 0

0 10]

Then the point (x0,y0) is a:

You flip a fair coin two times. What is the probability of getting one head and one tail in any order?

You throw two dice and sum the result, what is the probability the sum is equal to 10?

You throw a six-sided dice 10 times, summing the result in each throw. What is the probability that the sum of results is greater than 10?

Hint: Use the complement rule

In an experiment, there are 100 patients. After taking medicine, 50 people experienced a headache and 50 people experienced a fever. The doctors want to find the probability that a patient may experience a headache or fever.

Which of the following statements is true?

A software company conducted a test on their new platform by exposing their users to two versions of the same product.

Number of users that were given version A: 4000

Number of user that were given version B: 5000

Number of users that experienced a bug: 3000

Number of users with version B that experienced a bug: 1500

What is the probability that a user tested Version B, given they experienced a bug during testing?

In a bag of marbles, there are two disjoint events: AA represents selecting a red marble, and B represents selecting a blue marble. The probability of selecting a red marble is P(A)=​ $$\frac{1}{4}$$ , and the probability of selecting a blue marble is P(B)= $$\frac{1}{3}$$

What is the probability of selecting either a red or a blue marble, $$P\left(A\cup B\right)$$ , from the bag?

You throw 10 fair coins, what is the probability that coins do not result in all heads?

In a room, there are 200 people: 30 people only like soccer, 100 people only like basketball, and 70 people like both soccer and basketball.

What is the probability that a randomly selected person likes basketball given they like soccer?

Hint: Find P(B|S), where B is the event of liking basketball and S is the event of liking soccer.

Imagine there is a disease that impacts 1% of the population. Researchers devised a test so that people with the disease test positive 95% of the time. People who do not have the disease test negative 90% of the time. If an individual receives a positive test result for the disease, what is the probability that they truly have the disease or $$P\left(sick\left|test_{pos}\right|\right)$$ ?

Hint: In the description above, you were given P(sick), probability for true positive (or $$P\left(test_{pos}\left|sick\right|\right)$$ , and probability for true negative (or $$P\left(test_{neg}\left|ot\ sick\right|\right)$$ . Use this information to find P(not sick)and $$P\left(test_{pos}\left|not\ sick\right|\right)$$ .

Remember that Bayes’ theorem $$P\left(A\left|B\right|\right)=\frac{P\left(B\left|A\right|\right)\cdot P\left(A\right)}{P\left(B\right)}$$ . Also, remember that you may write $$P\left(B\right)=P\left(B\left|E\right|\right)\cdot P\left(E\right)+P\left(B\left|not\ E\right|\right)\cdot P\left(not\ E\right)$$ , where E is any event and not E=E’.

Which of the following are examples of continuous random variables? Select all that apply.

You roll a six-sided die 20 times and want to find the probability that the number 4 appears exactly 7 times. Which of the following equations correctly represents the probability distribution for this scenario?

Imagine you are tasked with modeling the heights of individuals in a diverse country. Which probability distribution would be most suitable for capturing

A taxi cab service analyzes the number of passengers in its daily rides. The table and graph below show the number of passengers, X, in a single taxi cab and the observed probabilities at a randomly selected time.

What is the probability that a randomly selected taxi ride will have less than or equal to 3 passengers?

Select the correct Cumulative Distribution Function based on the observed probabilities

Consider the graph below, depicting four normal, or Gaussian, distributions labelled normal_A in blue, normal_B in orange, normal_C in green, and normal_D in red.

Consider the four sets of samples above. Which one has the smallest variance?

Consider two games, Game A and Game B, each with different probability distributions of winnings and losses. Game A has a probability of $$\frac{1}{3}$$ to win $2 and a probability of $$\frac{2}{3}$$ to lose $1. Game B has a probability of $$\frac{1}{2}$$ ​ to win $0.50, a probability of $$\frac{1}{4}$$ to lose $0.50, a probability of $$\frac{1}{8}$$ to win $5, and a probability of $$\frac{1}{8}$$ ​ to lose $2.

Consider the following independent random variables:

X∼Normal(3, $$1^2$$ )

Y∼Normal(2, $$2^2$$ )

Then Z=X+Y∼Normal(μ,σ²), where μ,σ are equal to:

Which of the following statements is true?

Which of the following statements is true?

What is the expected mean E[X] for this probability distribution?

What is the advantage of looking at the standard deviation instead of the variance?

The box plot below shows the distribution of salaries for employees in three different company departments.

Based on the box plots above, which of the following statements are true? Select all that apply.

Which of the following QQ plots represents a set of data that is more likely normally distributed?

The violin plot and box plot below are visualizations of the same dataset. Based on the violin plot and box plot above, which of the following statements are true?

Suppose that the joint probability distribution of two random variables X and Y is given by the following table:

What is the probability that X and Y both take even values?

Which of the following statements are true regarding marginal and conditional distributions? Select all that apply.

Suppose that the joint probability distribution of two random variables X and Y is given by the following table:

What is the conditional distribution $$P\left(X=3\left|Y=1\right|\right)$$ ?

Which of the following statements regarding the correlation coefficient are true? Select all that apply.

Suppose that the joint probability distribution of two random variables X and Y is given by the following table:

What is the covariance between X and Y?

What is the difference between a sample and a population in statistics?

Let S be a random sample, where S={5,2,7,10}. Calculate the population variance for the sample set.

A researcher conducts a study by taking independent random samples. Assuming the experiment meets the conditions of the Law of Large Numbers, which sample mean is the closest to the value of the population mean?

Which of the following best describes the Central Limit Theorem?

Consider the following population, P, where P={1,1,3,5,10}

And the following sample, S, where S={1,3}

What is the value of the sample mean?

Which of the following statements about the Central Limit Theorem (CLT) is true?

Which of the following methods can be used to estimate a population's variance, mean, and proportion?

Suppose you have a dataset of points and want to fit a line to best represent the relationship between the variables. Which of the following statements about linear regression is true?

What is the purpose of regularization in machine learning?

Assume you have a dataset that generates a model M= $$4x^4+3x^2+1$$ that best fits the data. What is the L2 regularization error value for the model M?

Which of the following best describes the way "priors" are used in Bayesian statistics?

Consider the following sample drawn from a Normal Distribution with unknown mean and unknown variance.

S={−1,0,1,2}

Select the Normal Distribution that is most likely to have been generated from sample S.

Suppose you flip a coin 10 times and obtain 6 heads and 4 tails. What function needs to be maximized to find the maximum likelihood estimate of the probability of getting heads on a single coin toss? Let p be the probability of getting heads.

You have observations of the numbers [−1,2] and you want to determine which distribution they could have been sampled from. The first distribution is a normal distribution N(0,22) with a μ=0 and σ=2. The second distribution is a normal distribution N(1,12) where μ=1 and σ=1.

Assume two Bayesian statisticians find a coin on the street and are trying to determine the likelihood that this coin will land on heads when flipped.

Bayesian 1 strongly believes that most coins are fair and begins with priors that are heavily concentrated around P(H)=0.5

Bayesian 2 assumes they know nothing about coins and begins with uniform priors with equal likelihood of every P(H) between 0 and 1.

In order to update their beliefs, they flip the coin 10 times, and get 3 heads and 7 tails.

Which of the following is the most likely value of their MAP beliefs once they've accounted for this data?

Which of the statements about confidence intervals is true?

Hint: margin of error= $$z_{\frac{\alpha}{2}}\cdot\frac{\sigma}{\sqrt[]{n}}$$

You have a sample size of 20 from a population with unknown mean and standard deviation. You measured that the sample mean X=50 and the sample standard deviation is s=10. What expression describes the margin of error for a confidence level of 95%?

In statistical hypothesis testing, which of the following statements correctly defines Type I and Type II errors?

When conducting a hypothesis test, after defining the null hypothesis (H0​) and the alternative hypothesis (H1​), what are the general steps to decide whether to reject the null hypothesis? Select the correct sequence of steps.

Suppose you are conducting a hypothesis test to determine whether a new teaching method improves student performance.

The null hypothesis (H0​) states that the teaching method has no effect, while the alternative hypothesis

(H1​) suggests that the teaching method leads to higher student performance. You collect data from a sample of 50 students and calculate a test statistic of 1.98. The critical value at a significance level of 0.05 is 1.96. Should you reject the null hypothesis?

A company claims that their new energy drink decreases reaction times. To investigate this claim, a researcher conducts a hypothesis test using a sample of 40 participants. The average reaction time in the sample is 0.95 seconds, with a standard deviation of 0.12 seconds. The company states that the average reaction time without their energy drink is 1.05 seconds. The researcher wants to determine whether sufficient evidence supports the company's claim. Assuming a significance level of 0.05, what is the test statistic for this hypothesis test?

Based on the scenario in the previous question (question #7), which distribution would you use to find p-values for different levels of significance?

You notice that your six-sided die seems to favor the outcome six. You state the null hypothesis is that the die is fair, and the alternative hypothesis is that the die favors some outcomes. After conducting a hypothesis test by rolling the die 100 times, you determine that the p-value is 0.03. Which of the following conclusions is a correct interpretation of the p-value?

Which of the following scenarios should be analyzed as a two-sample t-test?

Researchers conducted a study and tested a random sample of 200 animals. Their research shows that 40 of the animals test positive for a disease. Calculate the margin of error for a 90% confidence level for the percentage of animals that carry the disease.

Hint:

margin of error = $$z_{\frac{\alpha}{2}}\cdot\sqrt[]{\frac{p\left(1-p\right)}{n}}$$

Consider the following system of equations in two variables.

x+3y=15

3x+12y=3

Check all the options that are true, given the system above.

Consider the following system of equations in two variables.

2x+y=5

4x+2y=10

Check all the options that are true, given the system above.

Consider the following matrix:

[3 1

6 2]

Its rank is: