ECON 2500 (B): Midterm Exam
October 30th, 2024 (Fall Semester)
You have 85 minutes to complete the test. There are 5 problems in the test, and each gives up to 5
points. The number of points for each particular sub-question is written next to it. Please, respect the
silence and raise your hand in case of a question.
1. [Types of variables, sample distributions, basic probability] Indicate whether the claim is True or
False, and (very briefly) explain why.
(a) [1 points] Variable “Total area (measured in square feet)” in the dataset with residential houses
is of categorical ordinal type.
(b) [1 points] Inter-quartile range is less sensitive to outlier observations than sample variance.
(c) [1 points] Distribution of monthly salaries across all employees of the Bank of Montreal is likely
left-skewed.
(d) [1 points] A normal random variable can take only non-negative values.
(e) [1 points] A mathematical concept of random experiment implies that only a single elementary
outcome realizes after the experiment.
2. [Characteristics of numerical data] Consider the following samples
3, −1, 6, 0, 2
and
1, 0, 4, −2, 2
(a) [4 points (1 for each item)] For each sample, find i) sample mean, ii) median, iii) inter-quartile
range, iv) sample variance.
(b) [1 points] Calculate the sample correlation.
3. [Probability Space] Suppose the final score of a hockey game between teams A and B is modeled as
a random variable.
(a) [1 points] How many elements are in the Sample Space, Ω?
(b) [2 points] Draw a scheme for the Sample Space, Ω. [Hint: you can use a two-dimensional array
for this]
(c) [2 points] Assuming that all elementary outcomes from Ω have positive probabilities, consider
the following events: A = {"game ends in a tie"}, B = {"team A scores at least once"}, C =
{"team B scores 0"}. Are these events disjoint? Analyze, whether it can be true that P (A) +
P (B) + P (C) = 1?
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4. [Law of Total Probability, Bayes’ Theorem] Suppose it is known that a half of new cars of a considered
model have a defect. This defect implies that the engine breaks within the first month with probability
10%. The remaining half of the cars is of high quality implying that the engine malfunction happens
with probability of only 1% within the first month. The two types of cars are indistinguishable
otherwise.
(a) [2 points] Construct a probability tree for the following random process: a new car is randomly
taken, then it is tested for one month and the problem with engine is either happens or not.
(b) [3 points] Assuming that no malfunction happens within a month, what is the probability that
the car is defected?
5. [Random Variables] Using historical data, a flight company found that when someone buys a flight
ticket there is a 5% probability that the person will not show up for the flight. An aircraft has 300
seats for passengers and the company sells 310 tickets for a flight. Let X be the number of people
who have bought these tickets and then do show up for the flight.
(a) [0.5 points] Use Binomial distribution to model X as a random variable. What are the values
of parameters n and p of such distribution?
(b) [1 point] Calculate E(X) and V (X).
(c) [1 point] Can we apply De Moivre-Laplace Theorem here? Explain.
(d) [0.5 points] If we approximate X using the normal distribution, what mean and variance does
it have?
(e) [2 points] Explain how would you find the probability that X > 300, i.e. probability that not
enough seats will be available. In your explanation, give as much details as possible.
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