Course: Mathematics-I (CDA101), Topic- Joint PMF, PDF, Covariance,
Correlation
Tutorial Sheet #5
1. A privately owned business operates both a drive-in 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-in and the walk-in facilities are in use, and suppose that the
joint density function of these random variables is
2
5
(2x + 3y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
f (x, y) =
0 : otherwise
Find P [(X, Y ) ∈ A], where A = {(x, y)|0 ≤ x ≤ 1/2, 1/4 ≤ y ≤ 1/2}.
2. Find marginal PDFs for random variable X and Y in above problem.
3. Two ballpoint pens are selected at random from a box that contains 3 blue pens, 2
red pens, and 3 green pens. If X is the number of blue pens selected and Y is the
number of red pens selected, find
(a) the joint probability mass function,
(b) P [(X, Y ) ∈ A], where A is the region {(x, y)|x + y ≤ 1}.
4. In above problem 3, find the conditional distribution of X, given that Y = 1, and
use it to determine P (X = 0|Y = 1).
5. Find the expected value of g(X, Y ) = XY in above problem 3.
6. Find the covariance of X and Y in problem 3.
7. Find the correlation coefficient between X and Y in problem 3.
8. The joint density for the random variables (X, Y ), where X is the unit temperature
change and Y is the proportion of spectrum shift that a certain atomic particle
produces, is
10xy 2 : 0 < x < y < 1
f (x, y) =
0 : otherwise
(a) Find the marginal densities g(x), h(y), and the conditional density f (y|x).
(b) Find the probability that the spectrum shifts more than half of the total obser-
vations, given that the temperature is increased by 0.25 unit.
9. Given the joint density function
x(1+3y2 )
: 0 < x < 2, 0 < y < 1
f (x, y) = 4
0 : otherwise
find g(x), h(y), f (x|y), and evaluate P (1/4 < X < 1/2|Y = 1/3).
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Correlation
Tutorial Sheet #5
1. A privately owned business operates both a drive-in 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-in and the walk-in facilities are in use, and suppose that the
joint density function of these random variables is
2
5
(2x + 3y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
f (x, y) =
0 : otherwise
Find P [(X, Y ) ∈ A], where A = {(x, y)|0 ≤ x ≤ 1/2, 1/4 ≤ y ≤ 1/2}.
2. Find marginal PDFs for random variable X and Y in above problem.
3. Two ballpoint pens are selected at random from a box that contains 3 blue pens, 2
red pens, and 3 green pens. If X is the number of blue pens selected and Y is the
number of red pens selected, find
(a) the joint probability mass function,
(b) P [(X, Y ) ∈ A], where A is the region {(x, y)|x + y ≤ 1}.
4. In above problem 3, find the conditional distribution of X, given that Y = 1, and
use it to determine P (X = 0|Y = 1).
5. Find the expected value of g(X, Y ) = XY in above problem 3.
6. Find the covariance of X and Y in problem 3.
7. Find the correlation coefficient between X and Y in problem 3.
8. The joint density for the random variables (X, Y ), where X is the unit temperature
change and Y is the proportion of spectrum shift that a certain atomic particle
produces, is
10xy 2 : 0 < x < y < 1
f (x, y) =
0 : otherwise
(a) Find the marginal densities g(x), h(y), and the conditional density f (y|x).
(b) Find the probability that the spectrum shifts more than half of the total obser-
vations, given that the temperature is increased by 0.25 unit.
9. Given the joint density function
x(1+3y2 )
: 0 < x < 2, 0 < y < 1
f (x, y) = 4
0 : otherwise
find g(x), h(y), f (x|y), and evaluate P (1/4 < X < 1/2|Y = 1/3).
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