STAT 3021FALL 2022
Midterm Exam (ALL)
Time Limit: 50 Minutes
Name (Print):
Instructions:
• Do not begin or turn this page until you are instructed.
• Enter all requested information on the top and bottom of this page, and put your
initials on the top of every page, in case the pages become separated.
• The exam has four problems I (20 pts), II (15 pts), III (35 pts) and IV (30 pts)
with subparts for each. The problem III will have 10 pts bonus.
• The exam is closed book. You may not use your books, or any wireless device on this exam.
• You may use a calculator and one sheet of paper (size A4 or 8.5” by 11”) with formulas or
other notes on both sides. You may not share calculators or notes!
• Show all your work on each problem for full credit. The following rules apply:
– Organize your work, in a reasonably neat and coherent way, in the space provided. Work
scattered all over the page without a clear ordering will receive very little credit.
– Mysterious or unsupported answers will not receive full credit for short answer problems.
A correct answer, unsupported by calculations, explanation, or algebraic work will not
receive full credit; an incorrect answer supported by substantially correct calculations
and explanation may still receive partial credit.
– If you need more space, use the back of the pages; clearly indicate when you have done
this.
Honesty Statement and Pledge:
I have not given or received any aid or assistance to or from any other student in this course during
the exam period. Everything I have written on this exam represents my own work and knowledge.
I sign this knowing that infringements on the University’s Academic Honest policy may result in
failure or expulsion.
Signed By:
Date:
STAT 3021, FALL 2022
Midterm Exam (ALL) – Page 2 of 8
Initials:
Problem I. (20 points) In a sample space S, consider three events A, B, C. Suppose P (A) =
0.3, P (B ′ ) = P (B c ) = 0.6 and P (C) = 0.5. If we know P (A ∪ C) = 0.6, P (B ∩ C) = 0.3 and
P (A ∩ B ∩ C) = 0.1. Do the following question. (Hint: You can draw a Venn diagram to help
you with this problem, though it’s not necessary).
1. (5 points) Find the probability P (A ∩ C).
2. (10 points) Find P (C ∩ (A ∪ B)). (Hint: Distribution Rule). Furthermore, find the
conditional probability P (A ∪ B|C).
STAT 3021, FALL 2022
Midterm Exam (ALL) – Page 3 of 8
Initials:
3. (5 points) Are A, B mutually exclusive? Why? (Reasoning with computation).
STAT 3021, FALL 2022
Midterm Exam (ALL) – Page 4 of 8
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Problem II. (15 points) Be sure to show all work for full credit.
In a standard (Poker) deck with 52 cards (without the two Jokers), if you randomly pick up 4
cards (without replacement),
I . find the probability that you EXACTLY get Two pairs(eg. 55JJ, but not 5555). (Hint:
4 different suits should be considered, and order does not matter) (10 pts)
II . find the probability of four of a kind, (eg. KKKK). (5 pts)
STAT 3021, FALL 2022
Midterm Exam (ALL) – Page 5 of 8
Initials:
Problem III. (35 points) Be sure to show all work for full credit.
Consider the joint pdf for random variable (X, Y ) as:
(
6x
x, y > 0 and x + y < 1
f (x, y) =
0
elsewhere
1. (10 points) Verify f is a valid density function.
2. (15 points) Find the marginal density of Y , fY (y) and the expectation of Y , E[Y ].
STAT 3021, FALL 2022
Midterm Exam (ALL) - Page 6 of 8
Initials:
3. (10 points) Find the conditional distribution X|Y density f (x|y) for every y ∈ (0, 1).
4. (10 points) (Bonus question): Find the expectation of X|Y , i.e. E[X|Y ] (This expectation
depends on Y ) (4 pts); Treat E[X|Y ] as a r.v. of Y , find the expectation E[E[X|Y ]] (4
pts); If we know E[X] = 1/2, how does E[E[X|Y ]] compare with E[X] (2 pts)?
STAT 3021, FALL 2022
Midterm Exam (ALL) - Page 7 of 8
Initials:
Problem IV. (30 points) Let X follow a binomial distribution Bin(10, 0.5) trial 10, success rate
1/2. Let Y follow a exponential distribution Exp(2) (The parameters are the same as defined
in slides). The covariance of X and Y is Cov(X, Y ) = 1.
1. (10 points) Find the expectation of X − Y , E[X − Y ].
2. (10 points) Find the correlation of X and Y , Corr(X, Y ) = ρXY .
3. (10 points) Find the E[X 2 + XY + Y 2 ].
STAT 3021, FALL 2022
Midterm Exam (ALL) - Page 8 of 8
Initials:
Please do NOT write in the following table. This is for grading purpose only!
Question
Score
I (20)
II (15)
III (35+10)
IV (30)
110
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