Name & NetID:
STAT 310 Exam 1
STAT 310 Exam 1
Monday, May 20, 2024
Name:
NetID:
Honor Statement:
Before turning in your exam, please sign this statement in the space below:
“On my honor, I have neither given nor received any unauthorized aid on this exam.”
If you do not complete this cover page, there will be a penalty of up to 5 points.
Instructions:
1. You have 75 minutes to complete the exam and an additional 5 minutes to upload your exam
to Gradescope. If you arrive late, you forfeit that time.
2. Show your work, and clearly mark or box your answers.
On free response problems, a “lonely” correct final answer (i.e., the correct answer but without
sufficient work shown) will not earn full credit. Convince me you know what you’re doing.
3. If you experience technical difficulties uploading your exam to Gradescope, email me your
exam. Send your email within the 80-minute exam window to avoid a points penalty.
4. You may use any course materials, such as notes, textbooks, homework solutions, etc.
5. You may not use online resources, such as Google, Stack Exchange, WolframAlpha, ChatGPT,
etc.
6. You must show complete integration and differentiation steps to earn full credit.
7. You may use a handheld scientific or a graphing calculator.
• No sharing calculators during the entire exam.
• You may not use a calculator for the following calculations: integration, differentiation,
normal distribution.
8. Absolutely no collaboration is allowed.
Even after you turn in the exam, you may not communicate with other people about it until
AFTER I post the exam solutions.
You may communicate with me (Instructor Ye) to ask a question during the exam.
Exam 1 ⋄ Page 1/8
Name & NetID:
STAT 310 Exam 1
Exam Problems
Problems 1-6 are multiple choice questions. For each question, darkly and completely bubble in your
single answer choice. No justification is required.
1. (3 points) Which of the following is NOT equivalent to P (A) + P (Ac )?
⃝ P (A ∩ Ac )
⃝ P (A ∪ Ac )
⃝ P (Ω)
⃝ P (B) + P (B c )
⃝ Not enough information
2. (3 points) Which of the following is true about conditional probability P (A | B)?
⃝ P (A | B) is always higher than P (A) because knowing B had happened gives us a better
chance
⃝ P (A | B) is always lower than P (A) because knowing B had happened means we should be
more conservative
⃝ If B = Ac , then P (A | B) must be 0
⃝ It is not possible to have P (A | B) = 0 or 1
⃝ None of the above
3. (3 points) During your exam study, you found that the practice exam solution was quite challenging
to understand, so you emailed the Instructor for clarification. While you were bored waiting for a
reply, you decided to model how long it takes the Instructor to get back to you. Which of the
following distributions is the most appropriate for your model?
⃝ Binomial
⃝ Geometric
⃝ Poisson
⃝ Uniform
⃝ Exponential
⃝ Normal
⃝ Not enough information
4. (3 points) Which of the following is true about variance?
⃝ Adding 100 to each possible value of X will change the variance
⃝ Dividing each possible value of X by π won’t change the variance
⃝ Var(2X) ≥ 4E[X 2 ]
⃝ Var(log(X)) = log(Var(X))
⃝ None of the above
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STAT 310 Exam 1
5. (3 points) Let X be a random variable and A = {1, 2, 3} be a subset of supp(X). That is, {1, 2, 3}
are possible values of X but it can take on more values. Which of the following is not true?
X
⃝ If X is discrete, then P (X ∈ A) =
p(x), where p(x) is the PMF of X
x∈A
Z 3
⃝ If X is continuous, then P (X ∈ A) =
f (x) dx, where f (x) is the PDF of X
1
⃝ If X is discrete, then P (X ∈ A) ̸= 0
⃝ If X is continuous, then P (X ∈ A) = 0
⃝ None of the above
Problems 6–11 are free response problems. For each question, show all work to convince me you
know what you’re doing.
6. (8 points) The historical mean for Exam 1 scores was 83 and your friend got 108! Say we want to
model Exam 1 scores with a normal distribution. For what standard deviation would your friend’s
score be in the top 2.5% of the class?
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STAT 310 Exam 1
7. Since 1980, 81 hurricanes affected Texas, about 1.84 hurricanes per year.
(a) (6 points) What is the probability of Texas being affected by 2 hurricanes next month?
(b) (8 points) What is the probability that the next hurricane is at least 2 months from now?
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STAT 310 Exam 1
8. (10 points) Imagine a game where there are 10 balls in an urn, numbered 1 through 10. The player
is to draw 3 balls out at random without replacement. If the 2nd largest number drawn is either 4
or 5, then the player wins the game. What is the probability of winning this game?
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STAT 310 Exam 1
9. (8 points) You are tasked to implement a device to alert Rice U personnel when heavy rain is
approaching in the next 24 hours. Through your clever engineering, you invented a device that sends
out alerts when it should with a probability of 0.97. However, it will also send out alerts when it
shouldn’t with a probability of 0.05. Based on historical data, the chance of observing heavy rain in
any 24-hour period is 2%. If I receive an alert from your device, what is the probability that there
will actually be heavy rain in the next 24 hours?
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STAT 310 Exam 1
10. (8 points) Let Y be a continuous random variable with PDF
f (y) =
ec
,
(y + 1)3
for y > 0.
Find c such that f (y) is a valid PDF.
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STAT 310 Exam 1
11. (9 points) The annual profit, W , in millions of dollars, of a tech company can be expressed as a
function of their product demand, X:
W = 2(1 − e−2X ).
Suppose that we can model X with an Exponential distribution with λ = 4. Find the expected
annual profit of this tech company.
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