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MLF week 10 - real exam questions
45 questions pulled from past papers whose wording matches this week's topics, newest first, duplicates removed. Pick your answers, then press Check answers. Figures and answer keys are the originals.
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Let 
be a discrete random variable such that 
with 
where 
is an unknown parameter. Consider the random sample 
from . The maximum likelihood estimate of 
is:
be a discrete random variable such that with where is an unknown parameter. Consider the random sample from . The maximum likelihood estimate of is:
Let 
where 
and 
are independent. Define 
Which of the following is the value of 
where and are independent. Define Which of the following is the value of 
A nonnegative random variable satisfies 
Using Markov's inequality, find the smallest value of 
such that 
Enter your answer as an integer.
satisfies Using Markov's inequality, find the smallest value of such that Enter your answer as an integer.Let 
be a real symmetric matrix. Consider the following statements. (1) is orthogonally diagonalizable. (2) is not diagonalizable. (3) All the eigenvalues of are real. (4) The eigenvalues of may be imaginary. (5) Eigenvectors corresponding to distinct eigenvalues of are linearly independent. (6) Eigenvectors corresponding to distinct eigenvalues of are linearly dependent. Which of the above statements are true?
be a real symmetric matrix. Consider the following statements. (1) is orthogonally diagonalizable. (2) is not diagonalizable. (3) All the eigenvalues of are real. (4) The eigenvalues of may be imaginary. (5) Eigenvectors corresponding to distinct eigenvalues of are linearly independent. (6) Eigenvectors corresponding to distinct eigenvalues of are linearly dependent. Which of the above statements are true?- (1), (3), (5)
- (2), (4), (6)
- (2), (3), (5)
- (1), (4), (6)
is a square matrix of order 
with the eigenvalue 
repeated twice. Which of the following is/are true?- is invertible.
- Each of the entries on the main diagonal of is equal to .
- has two linearly independent eigenvectors.
is not invertible, where 
is the identity matrix.
Suppose is a 
matrix with rank 4. Which of the following is/are true?
is a matrix with rank 4. Which of the following is/are true?- Columns of are linearly independent.
- Null space of contains only the zero vector.
- Row space of is
. - Left null space of has dimension 2.
Let 
. That is, 
is distributed uniformly over 
. Define 
. Which of the following is true regarding the random variable 
?
. That is, is distributed uniformly over . Define . Which of the following is true regarding the random variable ?
The number of automobiles sold weekly at a certain dealership is a random variable with expected value 
. Using Markov's inequality, find the tightest upper bound to the probability that the next week's sales is at least 
. Enter your answer correct to one decimal place. Note that an upper bound of 
is valid but not tight.
. Using Markov's inequality, find the tightest upper bound to the probability that the next week's sales is at least . Enter your answer correct to one decimal place. Note that an upper bound of is valid but not tight.Let be an exponentially distributed random variable with mean 
. Find the value of 
for which 
. Enter your answer correct to two decimal places.
be an exponentially distributed random variable with mean . Find the value of for which . Enter your answer correct to two decimal places.The joint density function of and is given by:
Compute 
. Enter your answer correct to two decimal places.
and is given by:
Compute . Enter your answer correct to two decimal places.The dataset 
is sampled from a distribution whose PDF is given below with parameter 
:
Assume that each point in the dataset is sampled independently from this distribution. Find the maximum likelihood estimate for . Your answer should be an integer.
is sampled from a distribution whose PDF is given below with parameter :
Assume that each point in the dataset is sampled independently from this distribution. Find the maximum likelihood estimate for . Your answer should be an integer.Aman is evaluating candidates for a Machine Learning role. The time (in hours) taken to complete a technical interview round follows the probability density function:
Find the expected time taken to complete an interview. Your answer should be an integer.
(in hours) taken to complete a technical interview round follows the probability density function:
Find the expected time taken to complete an interview. Your answer should be an integer.Let be the time (in minutes) a data scientist takes to preprocess a dataset before training a model. Suppose the expected preprocessing time is 40 minutes.Using Markov’s inequality, find an upper bound on the probability that the preprocessing time is at least 1.25 times its expected value.
be the time (in minutes) a data scientist takes to preprocess a dataset before training a model. Suppose the expected preprocessing time is 40 minutes.Using Markov’s inequality, find an upper bound on the probability that the preprocessing time is at least 1.25 times its expected value.
Comprehension passage for this question
Suppose the completion time (in minutes) of a coding assignment during a machine learning interview follows a continuous uniform distribution 
. Based on the above data, answer the given subquestions.What is the conditional probability that the assignment takes more than 
minutes given that it has already taken more than 
minutes 
minutes given that it has already taken more than minutes 
Let and be two random variables having joint density function:
Find the value of 
. Enter the answer correct to three decimal places.
and be two random variables having joint density function:
Find the value of . Enter the answer correct to three decimal places.Let be a discrete random variable such that 
and 
where is an unknown constant. Consider a random sample 
. Find the maximum likelihood estimate of for the given sample. Enter the answer correct to one decimal place.
be a discrete random variable such that and where is an unknown constant. Consider a random sample . Find the maximum likelihood estimate of for the given sample. Enter the answer correct to one decimal place.Comprehension passage for this question
Consider the following dataset in 
Run PCA on this dataset. Based on the above data, answer the given subquestions.If
is the covariance matrix of the dataset, find 
. Enter the nearest integer as your answer.
is the covariance matrix of the dataset, find . Enter the nearest integer as your answer.Comprehension passage for this question
Consider the following dataset in Run PCA on this dataset. Based on the above data, answer the given subquestions.Let 
be the unit-norm eigenvector corresponding to the smallest eigenvalue of the covariance matrix. Find the variance of the dataset along the direction . Enter the nearest integer as your answer.
be the unit-norm eigenvector corresponding to the smallest eigenvalue of the covariance matrix. Find the variance of the dataset along the direction . Enter the nearest integer as your answer.Let 
be a continuous random variable with the probability density function:
What is the value of 
be a continuous random variable with the probability density function:
What is the value of Comprehension passage for this question
A cafe records the time (in minutes) a customer waits for a freshly made sandwich. The waiting time follows an exponential distribution with mean 5 minutes. Based on the above data, answer the given subquestions.What is the probability that a randomly chosen customer waits between 
and 
minutes 
and minutes 
Comprehension passage for this question
A cafe records the time (in minutes) a customer waits for a freshly made sandwich. The waiting time follows an exponential distribution with mean 5 minutes. Based on the above data, answer the given subquestions.Using Chebyshev's inequality, find an upper bound for 
, where 
represents the mean of the random variable . Enter your answer correct to two decimal places.
, where represents the mean of the random variable . Enter your answer correct to two decimal places.Comprehension passage for this question
If 
and 
, where
Based on the above data, answer the given subquestions.Find the covariance matrix of 
.
.
An analyst monitors the number of defective items produced daily. Let denote number of defective items produced in a day. Using Markov's inequality, the analyst finds an upper bound of 
on the probability that at least 
items are defective. Calculate the mean of .
denote number of defective items produced in a day. Using Markov's inequality, the analyst finds an upper bound of on the probability that at least items are defective. Calculate the mean of .Consider two datasets in 
, 
and 
, each having three points as shown below. The figure is drawn to scale. PCA is run on both these datasets separately.
and 
are the covariance matrices of the two datasets respectively. If 
is the eigenvector corresponding to the largest eigenvalue of and 
is the eigenvector corresponding to the smallest eigenvalue of , find the absolute value of 
. Both eigenvectors have unit norm. Your answer should be an integer.
, and , each having three points as shown below. The figure is drawn to scale. PCA is run on both these datasets separately.
and are the covariance matrices of the two datasets respectively. If is the eigenvector corresponding to the largest eigenvalue of and is the eigenvector corresponding to the smallest eigenvalue of , find the absolute value of . Both eigenvectors have unit norm. Your answer should be an integer.Anil throws a dart onto a circular board. Let a random variable denote the distance from the center to the point where the dart hits the board. Suppose the PDF of is
Calculate the value of
. Enter the answer correct to two decimal places.
denote the distance from the center to the point where the dart hits the board. Suppose the PDF of is
Calculate the value of
. Enter the answer correct to two decimal places.Comprehension passage for this question
If and , where
Based on the above data, answer the given subquestions.Find the covariance matrix of .
.
Comprehension passage for this question
Based on the above data, answer the given subquestions.
The sample covariance matrix C for the given dataset is
Comprehension passage for this question
Based on the above data, answer the given subquestions.
The eigenvalues of the covariance matrix are
In the context of Principal Component Analysis (PCA), which of the following statements is/are correct?
- PCA projects the original data onto a lower-dimensional subspace thatcaptures the maximum variance.
- The eigenvectors of the covariance matrix correspond to the directions ofmaximum variance.
- The principal components form a basis that is not necessarily orthogonal.
- The principal components obtained through PCA are always uncorrelated.
In the context of Linear Algebra, which of the following statements about Principal Component Analysis (PCA) is/are true?
- The principal components of a matrix are given by the eigenvectors of itscovariance matrix.
- The principal components of a matrix are given by the singular values of itscovariance matrix.
- PCA finds the principal components by maximizing the variance along eachcomponent.
- The principal components form a basis that is not necessarily orthogonal.
Comprehension passage for this question
Based on the above data, answer the given subquestions.
Compute the sample covariance matrix C for the given dataset.
Suppose a random variable X has a mean μ of 70 and a standard deviation σ of 8. Using Chebyshev’s inequality, determine the maximum probability that X will deviate from the mean by more than 16 units. Enter the answer correct to two decimal places.
Which of the following is/are true about PCA?
- PCA will transform the original data set onto a lower dimension subspace suchthat the variance of the project is maximized.
- PCA calculates the mean of each data set to determine its significance.
- PCA can be used to reduce the dimensionality of the dataset.
- PCA will transform the original data set onto a lower dimension subspace suchthat the reconstruction error is maximized.
What is the primary goal of Principal Component Analysis (PCA)?
- To find the directions that capture the maximum variance in the data.
- To find directions that capture the minimum variance in the data.
- To increase the dimensionality of the data.
- To reduce the number of data points.
In a large town, it is estimated that 20% of students own a laptop. A random sample of 400 students is selected from the town. What is the variance of the number of students with the laptops in a given sample?
A quality control manager at a factory measures the thickness of glass sheets produced. Five randomly selected sheets have the thicknesses(in mm) 10, 8, 11, 9, 12 respectively. Assuming the thickness follows a normal distribution and the variance is known, what is the maximum likelihood estimator (MLE) of the mean thickness, (μ), in mm of the glass sheets?
Comprehension passage for this question
Which of the following represents the covariance matrix, C for the given data points?
Comprehension passage for this question
What is the value of the projected variance?
Comprehension passage for this question
What is the value of the projected variance? Enter the answer correct to two decimal places.
Which among the following statements are true about principal component analysis (PCA)?
- PCA is a supervised learning algorithm.
- Principal components are the eigenvectors of the sample covariance matrix.
- Principal components are eigenvectors of the centered data matrix.
- If p1 and p2 are the principal component vectors, then p1 is orthogonal to p2.
Which of the following statements accurately describes the key difference between classification and regression models?
- In classification, the output variable is continuous and real-valued, while inregression, it is categorical.
- Regression models are used for predicting probabilities, whereas classificationmodels focus on predicting absolute values.
- Classification models aim to find decision boundaries to separate data intoclasses, while regression models seek to predict a numeric value.
- In regression, the commonly used loss function is mean squared error, whilein classification, 0 -1 loss is typically employed.
Which among the following is true for PCA?
- The first principal component represents the direction along which thevariance of the dataset is maximized.
- The first principal component represents the direction along which thevariance of the dataset is minimized.
- The first principal component is the eigenvector corresponding to the smallesteigenvalue of the covariance matrix.
- The first principal component is the eigenvector corresponding to the largesteigenvalue of the covariance matrix.
Suppose a fair coin is tossed twice. Let A denote the event that the first toss is a tail. Let B denote the event that the second toss is a tail. Let C denote the event that out of the first two tosses, either both are heads or both are tails. Which among the following options are true? Select all that apply.
- A, B and C are independent.
- A, B and C are dependent.
- A, B and C are pairwise independent.
- P(A) = P(B) = P(C) = 1/2
The average life (in years) of an electric bulb follows exponential distribution with parameter λ = 0.4. Using Weak Law of large numbers, find a lower bound on the probability that the mean life of a random sample of 50 such bulbs falls between 1.5 and 3.5 years. Enter your answer correct to three decimal places.
What are the properties of principal components in PCA?
- All principal components are orthogonal to each other.
- The number of principal components for n-dimensional data are atmost n−1.
- The first principal component accounts for most of the possible variability ofthe original data i.e, maximum possible variance.
- The first principal component is the eigenvector of the covariance matrixcorresponding to the maximum eigenvalue.