SCHOOL OF BUSINESS, ECONOMICS AND MANAGEMENT
BAS210-Probability and Statistics
Tutorial
Due date: Wednesday 25 October 2023
Question 1
Let X be a discrete random variable with support S1 = {0, 1}, and let X be a discrete
random variable with support S2 = {0, 1, 2}. Suppose, in tabular form, that X and Y have
the following joint probability distribution f (x, y):
f(x,y)
X
0
1
fY (y)
0
Y
1
2
fX (x)
1
8
2
8
3
8
2
8
1
8
3
8
1
8
1
8
2
8
4
8
4
8
1
What is the conditional distribution of X given Y ? That is, what is g(x|y)?
Question 2
Let X be a discrete random variable with support S1 = {0, 1}, and let Y be a discrete
random variable with support S2 = {0, 1, 2}. Suppose, in tabular form, that X and Y have
the following joint probability distribution f (x, y):
f(x,y)
X
0
0
1/8
1
2/8
3/8
Y
1
2/8
1/8
3/8
Determine h(y|x)
Question 3
1
2 fX (x)
1/8
4/8
1/8
4/8
2/8
1
The two-dimensional random variables X, Y have the joint density
(
8xy,
0<x<y<1
f (x, y) =
0,
otherwise
(1) Find [P (X <
1
2
∩ Y < 41 )]
(2) Find the marginal and condtional distributions
(3) Are X and Y independent?
Question 4
Let X and Y be independent exponentially distributed random variables with common
density
f (x) = α−αx ,
where α > 0. Show that the pdf for Z = X + Y is given by
α2 ze−αz
Question 5
What is the probability desnsity of the two independent random variables each of which is
gamma with parameter α = 1 and β = 1?
Question 6
Suppose we choose two numbers at random from the interval [0, ∞] with an exponential
density with parameter λ. What is the density of their sum?
Question 7
Prove Cov(X, Y ) = Cov(Y, X).
Question 8
Let X and Y have joint density given by
(
2x,
f (x, y) =
0,
0 ≤ x ≤ 1, 0 ≤ y ≤ 1
elsewhere
Find the covariance of X and Y .
Question 9
If the coefficient of correlation between x and y is 0.5 and their covariance is 16, and the
σx = 4, then what is the σy ?
Question 10
2
The fraction X of male runners and the fraction Y of female runners who compete in
marathon races are described by the joint density function
(
8xy,
0 ≤ x ≤ 1, 0 ≤ y ≤ 1,
f (x, y) =
0,
elesewhere.
(a) Find the covariance of X and Y
(b) Find the correlation coefficient of X and Y
3