Problem Set - IV

Due Date - April 23, 2025

Instructions

• This problem set is to be submitted individually. However, you are permitted to

• Maximum possible points is 50. This problem set counts for 10% of your final course
grade.

• There are five sections to this problem set.

• All the answers to this problem set must be presented in a pdf document. The for-
mat for pdf document is single-spacing, 12pt Times New Roman font with 1-inch
margins on all sides. If working in a paper-pen mode, you can take screenshots of
your work and convert it into a pdf document.

• The deadline for submission is April 23, 2025 11:59 PM. All submissions must be
made on Moodle. Your final submission for the pdf document should be named
"firstname_lastname_ps3.pdf".

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I - Continuous Random Variables (1+1+1+2= 5 points)

A random variable has a density function given by


2x + c for 3 ≤ x ≤ 4

f (x) =
0 otherwise


(i) Find the value of the constant c which ensures that f is indeed a density function.

(ii) Determine P [X ≤ 3.5]

(iii) Determine P [3.25 ≤ X ≤ 3.75]

(iv) Find the mean and standard deviation of X

II - Uniform Distribution (5 +5 = 10 points)

1) An elevator in a building can arrive any time between 5 and 10 seconds.

(i) What is the probability density function?

(ii) What is the distribution function?

(iii) What is the probability that the elevator arrives before 6.5 seconds?

(iv) What is the probability that the elevator arrives after 8 seconds?

2) The amount of time a car needs to wait (in seconds) at a toll gate is uniformly distributed
as in the following figure.

(i) What is the probability that a car waits fewer than 50 seconds?

(ii) What is the expected waiting time?

(iii) Find Var[X] where X captures the waiting time of car.

(iv) A randomly chosen person who drives regularly through the toll gate will wait x
seconds 75% of the time. Find x.

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 Figure 1: Distribution of wait times

III - Normal Distribution (2 + 3 + 2 + 4 + 4 = 15 points)

1) The heights of a certain population of males are normally distributed with mean 165
cms and standard deviation 5 cms. Approximate the proportion of this population whose
height is less than (a) 155 cms (b) greater than 180 cms.

2) Let X be a normal random variable with expected value 15 and standard deviation 4.

(i) Find P (7 ≤ X ≤ 15)

(ii) Find P (X > 23)

(iii) Find P (X < 19)

3) Find

(i) E[X] if X is a normal random variable with standard deviation 4 and the probability
that X is greater than 16 is 0.84.

(ii) SD[X] if Variable X is a normal random variable with expected value 200 and the
probability that X is greater than 90 is 0.975.

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4) A student is applying for a PhD program in the US university. To secure a fully funded
admit, the student is required to score within the top 15% of the quantitative section of the
GRE exam. Suppose the total points possible on the quantitative section is 170 and the exam
is normally distributed with expected value of 157 and standard deviation 4, determine the
approximate score required from the student to secure a fully funded PhD offer from any
US university.

5) The placement process for a data analytics job proceeds in two stages. The first stage is
a written exam and the second stage is a personal interview. A candidate will make it to
the interview if they score within the top 2% of the written exam. Suppose the total points
possible on the written test is 300, and given that the written exam is normally distributed
with expected value of 200 and standard deviation of 30, determine the approximate score
required from the candidate to be called for a personal interview.

IV - Standard Normal Distribution (6 + 3 + 3 + 3 = 15 points)

1)

(i) Find P (|Z| > 1.65)

(ii) Find P (|Z| < 2.2)

(iii) Find P (−2.5 < Z < 1.5)

(iv) Find x if P (Z > x) = 0.25

(v) Find x if P (|Z| < x) = 0.99

(vi) Find to two decimal places z0.12

2) Suppose the time taken to solve a puzzle is normally distributed with mean 500 seconds
and standard deviation 125 seconds. The local school team is conducting a qualifying round
for all the students interested in joining the school puzzle team. Only the top 10% of the
puzzle solvers will be selected into the school team. What is the critical time above which
one will not make it to the school puzzle team?

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3) Suppose the test scores on the APU entrance exam is normally distributed with a mean
of 70 and a standard deviation of 15.

(i) What proportion of students scored between 60 and 85.

(ii) What proportion of students scored above 9

(iii) What proportion of students scored less than 50.

4) You are considering buying a new gaming headset that will last you for the next 5 years.
You have narrowed down your search to either a Razer V2 or Epos H6. Your research tells
you that the longevity of Razer V2 is normally distributed with a mean of 3.5 years and
standard deviation of 1 year. And the longevity of Epos H6 is normally distributed with a
mean of 4 years and standard deviation of 2 years. If all you care about is that the headset
purchased lasts at least 5 years, which one should you buy?

V - Percentiles of Normal Distribution (5 points)

Suppose the exam scores on the economics exam are approximately normally distributed.
If 15% of the students score below 65 and 10% above 90, then what fraction of the students
score (a) below 80 (b) Between 70 and 95?