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DSC 40A - Homework
function
DSC 40A - Homework 5
Write your solutions to the following problems by either typing them up or handwriting them on another
piece of paper. Homeworks are due to Gradescope by 11:00pm on the due date. You can use a slip day to
extend the deadline by 24 hours. Make sure to correctly assign pages to Gradescope when submitting.
Homework will be evaluated not only on the correctness of your answers, but on your ability to present
your ideas clearly and logically. You should always explain and justify your conclusions, using sound
reasoning. Your goal should be to convince the reader of your assertions. If a question does not require
explanation, it will be explicitly stated.
Homeworks should be written up and turned in by each student individually. You may talk to other
students in the class about the problems and discuss solution strategies, but you should not share any
written communication and you should not check answers with classmates. You can tell someone how to do
a homework problem, but you cannot show them how to do it.
For each problem you submit, you should cite your sources by including a list of names of other students
with whom you discussed the problem. Instructors do not need to be cited.
This homework will be graded out of 50 points. The point value of each problem or sub-problem is indicated
by the number of avocados shown.
Note: Problems 1 and 2 refer to a supplemental Jupyter Notebook, which can be found at this link.
Problem 1. Transformation Tuesday
The logistic function, also known as the “sigmoid” function, is defined as follows:
σ(x) =
1
1 + e−x
The logistic function σ (has nothing to do with standard deviation) is used in a variety of fields. Pertinently,
it is used to model the growth of populations and spread of diseases. You’ll also see it later on in your data
science career when you learn about logistic regression.
a) Show that the inverse of the logistic function is given by
σ−1(x) = log
(
x
1− x
)
where log represents the natural logarithm with base e.
Hint: Recall, one strategy to find the inverse of a function y = f(x) is to write x = f ′(y) and solve
for y.
b) Note: Parts (b), (c), and (d) of this question should not take very much time; you’ve already
done the heavy lifting in part (a).
Suppose we have a dataset (x1, y1), (x2, y2), ..., (xn, yn) and want to use least squares to fit a prediction
rule
H(x) = σ(w0 + w1x)
1
This is not linear in our parameters, w0 and w1. However, through a transformation, we can frame
it as a linear prediction rule.
Using the process from Lecture 10, transform H(x) into a prediction rule that is linear in terms of the
parameters w0 and w1. Specify a design matrix X and observation vector z⃗ such that the optimal w
∗
0
and w∗1 are given by the solution to the normal equations X
TXw⃗∗ = XT z⃗. Your answers for X and
z⃗ may involve xi’s, yi’s, σ(·), and/or σ−1(·).
c) In the supplemental Jupyter Notebook, linked here, use the provided code and dataset to define
the design matrix and observation vector you specified in the previous part and to find w∗0 and w
∗
1 for
the prediction rule H(x) = σ(w0 + w1x). In your PDF writeup, provide a screenshot of the code you
wrote as well as of the resulting visualization.
d) As you saw in the supplemental Jupyter Notebook in the previous part, our prediction rule was
a good fit to our data.
What issue would arise using this technique if there were points in our dataset such that yi = 0 or
yi = 1?
Problem 2. What do you k-mean?
a) Consider the six data points given below, x⃗1 through x⃗5.
x⃗1 =
[
3
5
]
, x⃗2 =
[
7
20
]
, x⃗3 =
[
1
6
]
, x⃗4 =
[
2
4
]
, x⃗5 =
[
9
18
]
, x⃗6 =
[
10
21
]
Just by looking at the data, you should be able to roughly identify two clusters. Let’s see how k-means
clustering finds these clusters algorithmically.
Using x⃗1 and x⃗2 as initial centroids, trace through one iteration of the k-means clustering algorithm
by hand. What are the two centroids and what are the two clusters found after this first iteration?
b) In the supplemental Jupyter Notebook, linked here, you will find a walkthrough of using
k-Means Clustering on 209-dimensional data involving countries around the world. At the bottom of
that notebook you will find two questions; write the answers to those questions here.
Problem 3. License Plates
In this problem, we will examine general license plates from Texas, home to Billy the avocado farmer. In
Texas, license plates generally consist of 3 letters followed by 4 numbers. All letters are uppercase, and
repeated characters are allowed.
ABC-1234 is an example of an Texas license plate.
a) What is the probability that two randomly generated license plates match? You may leave
your answer as a product of powers of fractions.
b) What is the probability that three randomly generated license plates match? You may leave
your answer as a product of powers of fractions.
c) What is the probability that a randomly generated license plate begins with a vowel?
d) What is the probability that a randomly generated license plate begins with a vowel or ends
in a number divisible by 3? Simplify your answer.
2
Problem 4. Nine Lives
In this question, we will consider two fair 9-sided dice, each with faces numbered 1, 2, 3, ..., 9.
a) Suppose you roll the two dice and look at just one of them. You see that it’s an 8. What is the
probability that the sum of the two die rolls is 16?
b) Suppose you roll the two dice and look at both of them. You see that at least one of them
is a 5. What is the probability that the sum of the two die rolls is 9?
c) Suppose you roll the two dice and look at both of them. You see that exactly one of them
is a 5. What is the probability that the sum of the two dice rolls is 9?
Hint: it is not your answer to part (a) or part (b).
d) Suppose you roll the two dice and look at one of them. You see that this one die is less than
3. What is the probability that the sum of the two dice rolls is greater than 10?
Problem 5. Probability Rules for Three Events
a) The multiplication rule for two events says
P (A ∩B) = P (A) · P (B|A)
Use the multiplication rule for two events to prove the multiplication rule for three events:
P (A ∩B ∩ C) = P (A) · P (B|A) · P (C|(A ∩B))
On proving the above equation, can you identify a general trend in this methodology. For example,
what would be the multiplication rule for n events:
P (E1 ∩ E2 ∩ E3 ∩ ... ∩ En)
You do not need to spend too much time on this question. Your final answer should look similar to
the result of the previous part.
Hint: If E and F are two events, E∩F is also an event. Also, intersections/“and”s are “associative”,
meaning that E ∩ F ∩G = (E ∩ F ) ∩G = E ∩ (F ∩G); the same applies for unions/“or”s.
b) The general addition rule for any two events says:
P (A ∪B) = P (A) + P (B)− P (A ∩B)
Use the general addition rule for two events to prove the general addition rule for three events:
P (A ∪B ∪ C) = P (A) + P (B) + P (C)− P (A ∩B)− P (A ∩ C)− P (B ∩ C) + P (A ∩B ∩ C)
Some hints and guidance:
• While it’s a great idea to draw Venn diagrams to reason to yourself why this property holds true,
we are looking for an algebraic proof here, not a visual derivation.
• At some point, you may need to use the fact that if E, F , and G are events, then (E ∪ F ) ∩G =
(E ∩ G) ∪ (F ∩ G). Intuitively, the relationship between ∩ and ∪ is similar to the relationship
between multiplication and addition; if e, f, g are numbers, then (e+ f) · g = e · g + f · g as well.
c) To identify what students find most important in DSC 10, we want to administer a survey to
the students in DSC 20, DSC 30, and DSC 40A. Consider the following information:
3
• There are 300 students taking at least one of DSC 20, DSC 30, or DSC 40A right now.
• 200 students are taking DSC 20 right now, and 50 students taking DSC 30 right now. There are
no students taking both DSC 20 and DSC 30 right now.
• 50 students are taking both DSC 20 and DSC 40A right now, and 30 students are taking both
DSC 30 and DSC 40A right now.
Suppose I choose a single student uniformly at random from the population of students taking at least
one of DSC 20, DSC 30, and DSC 40A. What is the probability that they are enrolled in DSC 40A?
Simplify your answer.
Hint: Use the result in part (b).
Write your solutions to the following problems by either typing them up or handwriting them on another
piece of paper. Homeworks are due to Gradescope by 11:00pm on the due date. You can use a slip day to
extend the deadline by 24 hours. Make sure to correctly assign pages to Gradescope when submitting.
Homework will be evaluated not only on the correctness of your answers, but on your ability to present
your ideas clearly and logically. You should always explain and justify your conclusions, using sound
reasoning. Your goal should be to convince the reader of your assertions. If a question does not require
explanation, it will be explicitly stated.
Homeworks should be written up and turned in by each student individually. You may talk to other
students in the class about the problems and discuss solution strategies, but you should not share any
written communication and you should not check answers with classmates. You can tell someone how to do
a homework problem, but you cannot show them how to do it.
For each problem you submit, you should cite your sources by including a list of names of other students
with whom you discussed the problem. Instructors do not need to be cited.
This homework will be graded out of 50 points. The point value of each problem or sub-problem is indicated
by the number of avocados shown.
Note: Problems 1 and 2 refer to a supplemental Jupyter Notebook, which can be found at this link.
Problem 1. Transformation Tuesday
The logistic function, also known as the “sigmoid” function, is defined as follows:
σ(x) =
1
1 + e−x
The logistic function σ (has nothing to do with standard deviation) is used in a variety of fields. Pertinently,
it is used to model the growth of populations and spread of diseases. You’ll also see it later on in your data
science career when you learn about logistic regression.
a) Show that the inverse of the logistic function is given by
σ−1(x) = log
(
x
1− x
)
where log represents the natural logarithm with base e.
Hint: Recall, one strategy to find the inverse of a function y = f(x) is to write x = f ′(y) and solve
for y.
b) Note: Parts (b), (c), and (d) of this question should not take very much time; you’ve already
done the heavy lifting in part (a).
Suppose we have a dataset (x1, y1), (x2, y2), ..., (xn, yn) and want to use least squares to fit a prediction
rule
H(x) = σ(w0 + w1x)
1
This is not linear in our parameters, w0 and w1. However, through a transformation, we can frame
it as a linear prediction rule.
Using the process from Lecture 10, transform H(x) into a prediction rule that is linear in terms of the
parameters w0 and w1. Specify a design matrix X and observation vector z⃗ such that the optimal w
∗
0
and w∗1 are given by the solution to the normal equations X
TXw⃗∗ = XT z⃗. Your answers for X and
z⃗ may involve xi’s, yi’s, σ(·), and/or σ−1(·).
c) In the supplemental Jupyter Notebook, linked here, use the provided code and dataset to define
the design matrix and observation vector you specified in the previous part and to find w∗0 and w
∗
1 for
the prediction rule H(x) = σ(w0 + w1x). In your PDF writeup, provide a screenshot of the code you
wrote as well as of the resulting visualization.
d) As you saw in the supplemental Jupyter Notebook in the previous part, our prediction rule was
a good fit to our data.
What issue would arise using this technique if there were points in our dataset such that yi = 0 or
yi = 1?
Problem 2. What do you k-mean?
a) Consider the six data points given below, x⃗1 through x⃗5.
x⃗1 =
[
3
5
]
, x⃗2 =
[
7
20
]
, x⃗3 =
[
1
6
]
, x⃗4 =
[
2
4
]
, x⃗5 =
[
9
18
]
, x⃗6 =
[
10
21
]
Just by looking at the data, you should be able to roughly identify two clusters. Let’s see how k-means
clustering finds these clusters algorithmically.
Using x⃗1 and x⃗2 as initial centroids, trace through one iteration of the k-means clustering algorithm
by hand. What are the two centroids and what are the two clusters found after this first iteration?
b) In the supplemental Jupyter Notebook, linked here, you will find a walkthrough of using
k-Means Clustering on 209-dimensional data involving countries around the world. At the bottom of
that notebook you will find two questions; write the answers to those questions here.
Problem 3. License Plates
In this problem, we will examine general license plates from Texas, home to Billy the avocado farmer. In
Texas, license plates generally consist of 3 letters followed by 4 numbers. All letters are uppercase, and
repeated characters are allowed.
ABC-1234 is an example of an Texas license plate.
a) What is the probability that two randomly generated license plates match? You may leave
your answer as a product of powers of fractions.
b) What is the probability that three randomly generated license plates match? You may leave
your answer as a product of powers of fractions.
c) What is the probability that a randomly generated license plate begins with a vowel?
d) What is the probability that a randomly generated license plate begins with a vowel or ends
in a number divisible by 3? Simplify your answer.
2
Problem 4. Nine Lives
In this question, we will consider two fair 9-sided dice, each with faces numbered 1, 2, 3, ..., 9.
a) Suppose you roll the two dice and look at just one of them. You see that it’s an 8. What is the
probability that the sum of the two die rolls is 16?
b) Suppose you roll the two dice and look at both of them. You see that at least one of them
is a 5. What is the probability that the sum of the two die rolls is 9?
c) Suppose you roll the two dice and look at both of them. You see that exactly one of them
is a 5. What is the probability that the sum of the two dice rolls is 9?
Hint: it is not your answer to part (a) or part (b).
d) Suppose you roll the two dice and look at one of them. You see that this one die is less than
3. What is the probability that the sum of the two dice rolls is greater than 10?
Problem 5. Probability Rules for Three Events
a) The multiplication rule for two events says
P (A ∩B) = P (A) · P (B|A)
Use the multiplication rule for two events to prove the multiplication rule for three events:
P (A ∩B ∩ C) = P (A) · P (B|A) · P (C|(A ∩B))
On proving the above equation, can you identify a general trend in this methodology. For example,
what would be the multiplication rule for n events:
P (E1 ∩ E2 ∩ E3 ∩ ... ∩ En)
You do not need to spend too much time on this question. Your final answer should look similar to
the result of the previous part.
Hint: If E and F are two events, E∩F is also an event. Also, intersections/“and”s are “associative”,
meaning that E ∩ F ∩G = (E ∩ F ) ∩G = E ∩ (F ∩G); the same applies for unions/“or”s.
b) The general addition rule for any two events says:
P (A ∪B) = P (A) + P (B)− P (A ∩B)
Use the general addition rule for two events to prove the general addition rule for three events:
P (A ∪B ∪ C) = P (A) + P (B) + P (C)− P (A ∩B)− P (A ∩ C)− P (B ∩ C) + P (A ∩B ∩ C)
Some hints and guidance:
• While it’s a great idea to draw Venn diagrams to reason to yourself why this property holds true,
we are looking for an algebraic proof here, not a visual derivation.
• At some point, you may need to use the fact that if E, F , and G are events, then (E ∪ F ) ∩G =
(E ∩ G) ∪ (F ∩ G). Intuitively, the relationship between ∩ and ∪ is similar to the relationship
between multiplication and addition; if e, f, g are numbers, then (e+ f) · g = e · g + f · g as well.
c) To identify what students find most important in DSC 10, we want to administer a survey to
the students in DSC 20, DSC 30, and DSC 40A. Consider the following information:
3
• There are 300 students taking at least one of DSC 20, DSC 30, or DSC 40A right now.
• 200 students are taking DSC 20 right now, and 50 students taking DSC 30 right now. There are
no students taking both DSC 20 and DSC 30 right now.
• 50 students are taking both DSC 20 and DSC 40A right now, and 30 students are taking both
DSC 30 and DSC 40A right now.
Suppose I choose a single student uniformly at random from the population of students taking at least
one of DSC 20, DSC 30, and DSC 40A. What is the probability that they are enrolled in DSC 40A?
Simplify your answer.
Hint: Use the result in part (b).
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