University of Toronto
Department of Mechanical and Industrial Engineering
MIE236: Probability
(Fall 2022)
Tutorial 4: Joint distributions
Question 1: If the joint distribution of 𝑋 and 𝑌 is given by
𝑓(𝑥, 𝑦) =
(a)
(b)
(c)
(d)
𝑥 2𝑦+1
,
62
for 𝑥 = 0, 1, 2 and 𝑦 = 1, 2, 3, 4, draw out the distribution table and find:
𝑃(𝑋 < 2, 𝑌 = 2)
𝑃(𝑋 < 2, 𝑌 ≤ 2)
𝑃(𝑋 < 𝑌)
𝑃(𝑋 + 𝑌 = 3)
Question 2: A fast-food restaurant operates both a drive through facility and a walk-in facility. On a
randomly selected day, let 𝑋 and 𝑌, respectively, be the proportions of the time that the drive-through
and walk-in facilities are in use, and suppose that the joint density function of these random variables is
2
(𝑥
𝑓(𝑥, 𝑦) = {3 + 2𝑦),
0,
0 ≤ 𝑥 ≤ 1, 0 ≤ 𝑦 ≤ 1
𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
(a) Find the marginal density of 𝑋.
(b) Find the marginal density of 𝑌.
(c) Find the probability that the drive-through facility is busy less than one-half of the time.
Question 3: The joint probability density function of the random variables 𝑋, 𝑌, and 𝑍 is
4𝑥𝑦𝑧 2
𝑓(𝑥, 𝑦, 𝑧) = { 9 , 0 < 𝑥, 𝑦 < 1, 0 < 𝑧 < 3
0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
Find:
(a) the joint marginal density function of 𝑌 and 𝑍;
(b) the marginal density of 𝑌;
1
1
4
2
1
|
2
1
(c) 𝑃( < 𝑋 < , 𝑌 > , 1 < 𝑍 < 2);
(d) 𝑃(0 < 𝑋 <
3
1
𝑌 = 4 , 𝑍 = 2).
University of Toronto
Department of Mechanical and Industrial Engineering
MIE236: Probability
(Fall 2022)
Question 4: Consider the following joint probability density function of the random variables 𝑋 and 𝑌:
𝑓(𝑥, 𝑦) = {
3𝑥 − 𝑦
,
9
1 < 𝑥 < 3, 1 < 𝑦 < 2,
0,
𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
(a) Find the marginal density functions of 𝑋 and 𝑌
(b) Are 𝑋 and 𝑌 independent?
(c) Find 𝑃(𝑋 > 2).
Question 5: A small college has 90 male and 30 female professors. An ad hoc committee of five is
selected at random to write the vision and mission of the college. Let X and Y be the number of men and
women on this committee, respectively.
Find the joint probability mass function of X and Y.