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.

3. Upload your solutions to Gradescope by May 13, 10:00am PST.

4. Please ensure that:

— all uploaded files are clearly readable

— 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.

Academic honesty

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.

2. Sign your name.

3. Upload this document to Gradescope together with your solutions.

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

selected in the first step. Fully justify your answer.

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 N?1, . . . , N?n be

the rounded versions, where we either round up or down to the nearest integer. For

instance N?1 = 1 if N1 < 1.5 and N?2 = 17 if N2 ≥ 16.5. Let S =Pn

i=1 Ni be the

sum of the numbers and S? =Pni=1 N?i the sum of the rounded numbers. Using the

Central Limit Theorem, estimate the probability that |S ? S?| ≤ 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 ? N?i. How are they distributed? Notice that

S ? S? =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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