Final Examination - 11 AM
STAT 302 - Introduction to Probability
Winter Term I (2020–2021)
Final Exam Instructions
• The Final Exam must be completed individually. You are not allowed to receive any external help.
• To obtain full marks on a problem, you must provide all the steps required to arrive at the solution.
Ambiguous notations may lead to mark deductions. Show all the steps in your work as partial marks
will be given.
Problem 1 [12 marks]
In a casino game, 40 participants are randomly sold one ticket (one per participant). There are four
tickets that provide a prize. If the STAT 302 Teaching Team (comprised of Anthony, Ruben, Ian and
Jim) are participants in the game, what is the probability they win:
(a) exactly two of the prizes?
(b) none of the prizes?
You may leave your answers as an expression without evaluating their numerical values.
Problem 2 [12 marks]
Widgets produced by a factory have a 0.08 probability of being defective. Before being shipped off,
the widgets are independently inspected by two inspectors. The first inspector has a 0.9 probability of
detecting defective widgets, and the second inspector has a probability of 0.85 of detecting a defective
widget. Non-defective widgets are never flagged as defective by the inspectors.
(a) What is the probability that both inspectors do not detect a defective widget?
(b) What is the probability that a defective widget gets detected by at least one inspector?
(c) What is the probability that a widget is non-defective given that it was not flagged as defective by
the inspectors?
Problem 3 [12 marks]
An office in Vancouver just bought 5 monitors on Amazon from a company, but 15% of the monitors
sold by the company are defective. The repair cost to the company for the monitors is given by 2X + 5,
where X is the number of defective monitors. Find the variance of the cost.
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Problem 4 [12 marks]
A set of 20 electrical wires, each 10 meters long, must be painted. A point is independently selected on
each of the wires. The location of the point U us Unif(0, 10). If a wire’s point is within 2 meters of either
of the edges, it will be painted in red for \$1, otherwise in blue for \$2. What is the variance of the cost
to paint all 20 wires?
Problem 5 [16 marks]
The joint density function for random variables Y1 and Y2 is given by
f(y1, y2) =
{
c y1y
2
2 , 0 ≤ y1 ≤ 1, 0 ≤ y2 ≤ 2,
0, otherwise.
(a) Find the constant c such that f(x1, x2) is a valid density function.
(b) Compute E(Y1|Y2 = 1).
Problem 6 [12 marks]
Let Y |U = u ∼ Binom(N, u), where U ∼ Unif (0, 12). Compute Var(3Y + 2).
Problem 7 [12 marks]
Suppose that the moment generating function of the discrete random variable X is given by
MX(t) =
(1 + 4et) (1 + 5et)
30
.
Compute the variance of X, Var(X).
Problem 8 [12 marks]
Computers manufactured by a Vancouver company have a launch time that is a random variable with
mean 1.5 minutes and standard deviation 1 minute.
(a) Using Chebyshev’s Inequality, how many computers must be sold such that the average launch time
of the sold computers is within 0.15 minutes of the true mean with probability at least 0.95?
(b) Using the Central Limit Theorem, how many computers must be sold such that the average launch
time of the sold computers is within 0.15 minutes of the true mean with probability at least 0.95?
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