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Stat 134 Final Exam
Spring 2020
May 12, 10:00am PST
Instructions
1. There are 8 questions and a total of 40 marks. Attempt as many as you can.
2. Show all of your work and fully explain your reasoning. Cite any results from the
textbook or lectures that you use.
— every part of every question is on a separate page
— all files are in the proper order.
5. If DSP, you may instead email your solution to your GSI by May 13, 10:00pm PST if
150% time or by May 14, 10:00am PST if 200% time.
6. You MUST upload a declaration of academic honestly, as described below. Your
exam will not be accepted otherwise.
Your exam will be accepted ONLY IF you include a declaration of academic honesty.
1. In your own writing, copy out ALL of the following statements:
— As a member of the UC Berkeley community, I act with honesty, integrity, and respect
for others.
— I will not communicate with anyone about the exam, besides the instructor and GSIs
for the entire duration of the exam period.
— I will not refer to any books, notes, or online sources of information while taking the
exam, other than the course textbook, lecture notes and other materials available on
the official STAT 134 webpages.
— I will not take screenshots, photos or otherwise make copies of exam questions.
Page 1 of 3
1. [2 marks] Suppose that X and Y are independent standard normal random variables.
Using rotational symmetry, show that P(Y > √
3|X|) = 1/6.
Hint: Recall arctan(√
3) = π/3.
2. [5 marks] Every day a professor leaves their home in the morning and walks to their
office. Every evening they walk home. They take their umbrella with them only if it is
raining. If it is raining and they do not have their umbrella with them (at their home
or office), then they must walk in the rain. Suppose that it rains with probability
1/3 at the beginning of any given trip independently of all other trips. Show that
63/16 ≈ 4 is the expected number of days until the professor must walk in the rain
without their umbrella (either that morning or evening), supposing that initially they
have their umbrella with them at home.
Hint: Let µ be the expected number of days supposing they initially have their umbrella
with them at home, and let ν be the expected number of days supposing that they do
not. Explain why
and then, similarly, find an equation for ν in terms of µ and µ. Use these equations to
solve for µ.
3. In a large class, the midterm and final exam scores (M, F) are approximately bivariate
normal with µM = 60, σM = 25, µF = 65, σF = 20 and ρ =√3/2.
(a) [2 marks] Explain why 65 + 4ρ the expected final exam score of a student whose
midterm score is 65.
(b) [3 marks] Using Φ(1) ≈ 0.84, find the conditional probability that this student’s
final exam score is at least 75+4ρ (that is, at least 10 points higher than expected)
given that their midterm score is 65.
4. Cars arrive at a toll booth according to a rate λ Poisson process {Xt
: t ≥ 0}. Recall
that Xt
is the number of cars that have arrived by time t.
(a) [1 mark] Find the expected amount of time for the third car to arrive.
(b) [2 marks] With what probability does it take at least 4 units of time for the third
car to arrive?
(c) [2 marks] Suppose that buses also arrive at the same toll booth according to an
independent rate µ Poisson process {Yt
: t ≥ 0}. Using properties of exponential
random variables, explain why the inter-arrival times between visits by vehicles
(cars or buses) to the toll both are independent and identically distributed as
rate µ + ν exponential random variables.
Note: This follows by the super-position property, however this question is
essentially asking you to explain why this property holds.
Page 2 of 3
5. [5 marks] Recall Pólya’s Urn: Initially there is 1 red and 1 blue ball in an urn. In
each step, we select a ball from the urn uniformly at random, and then put it back
together with a new ball of the same color. Therefore after n steps, there are n + 2
balls in the urn. Suppose that after n steps there are r + 1 red and b + 1 blue balls,
where r + b = n. Show that r/(r + b) is the conditional probability that a red ball was
Hint: Let C1 be the color of the first ball selected. Let Rn the number of red balls
after n steps. Explain why
P(Rn = r|C1 = R) = 2n − 1r − 1!r!b!(n + 1)!
and
P(Rn = r|C1 = B) = b
r
P(Rn = r|C1 = R).
Argue directly, rather than by induction. Finally, apply Bayes’ Rule.
6. [5 marks] Suppose we generate n = 100 independent and identically distributed
uniformly random numbers N1, . . . , Nn on the interval [0, 100]. Let Nˆ1, . . . , Nˆn be
the rounded versions, where we either round up or down to the nearest integer. For
instance Nˆ1 = 1 if N1 < 1.5 and Nˆ2 = 17 if N2 ≥ 16.5. Let S =Pn
i=1 Ni be the
sum of the numbers and Sˆ =Pni=1 Nˆi the sum of the rounded numbers. Using the
Central Limit Theorem, estimate the probability that |S − Sˆ| ≤ 1. Use the value
Φ(√3/5) ≈ 0.64 to simplify your final answer as much as possible.
Hint: Consider the “round-offs” Ri = Ni − Nˆi. How are they distributed? Notice that
S − Sˆ =Pni=1 Ri.
7. This question has three unrelated parts:
(a) [3 marks] Let Θ ∼ Uniform(−π/2, π/2) and C = tan Θ. Find the probability
density function of C and identify its distribution.
(b) [3 marks] Suppose that U, V are independent, where U ∼ Uniform(0, 1) and
V ∼ Exponential(λ). Find the probability density function of the sum W = U +V .
(c) [4 marks] Let X1, . . . , Xn be an independent and identically distributed sequence
of Exponential(1) random variables, where n ≥ 3. Find the conditional probability
density function for the maximum M = X(n) given that the second smallest X(2) =
1. Use this to verify that, in the case that n = 4, E(M|X(2) = 1) = e
2/2 − 2e.
Hint: Start by using the “PDF method” to find the density for X(2) and the joint
density for (M, X(2)).
8. [3 marks] Suppose that there are n guests at a Halloween party, and that each is
wearing one of 200 possible costumes available at local store, uniformly at random
and independently of all other guests. Using Poisson approximation and the value

log 2 ≈ 0.83, show that only about n = 17 guests are needed to ensure that some
pair of guests are wearing the same costume with probabiliy at least 50%.
Note: This approximation is quite accurate. It can be shown that, when n = 17, the
true probability of a match is 1 − (200)17/20017 ≈ 50.3%.

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