BIVARIATE (JOINTLY) DISTRIBUTED DISCRETE RANDOM VARIABLE

HOMEWORK 5

1. Shown below is the joint probability distribution for two random variables X and Y.
X Y
5 10
10 0.12 0.08 0.20
20 0.30 0.20 0.50
30 0.18 0.12 0.30
0.60 0.40 1.00

a) Find , , and .
b) Specify the marginal probability distributions for X and Y.
c) Compute the mean and variance for X and Y.
d) Are X and Y independent random variables? Justify your answer.
2. There is a relationship between the number of lines in a newspaper advertisement for an apartment and the
volume of interest from the potential renters. Let volume of interest be denoted by the random variable X,
with the value 0 for little interest, 1 for moderate interest, and 2 for heavy interest. Let Y be the number of
lines in a newspaper. Their joint probabilities are shown in the table

Number of Volume of interest (X)

lines (Y) 0 1 2
3 0.09 0.14 0.07
4 0.07 0.23 0.16
5 0.03 0.10 0.11

a) Find and interpret .

b) Find the joint cumulative probability function at X=2, Y=4,
and interpret your result.
c) Find and interpret the conditional probability function for Y,
given X=0.
d) Find and interpret the conditional probability function for X,
given Y=4.
e) If the randomly selected advertisement contains 5 lines, what is the probability that it has heavy interest
from the potential renters?
f) Find expected number of volume of interest.
g) Find and interpret covariance between X and Y.
h) Are the number of lines in the advertisement and volume of interest independent of one another?
3. Students at a university were classified according to the years at the university (X) and number of visits to
a museum in the last year.
(Y=0 for no visits, 1 for one visit, 2 for two visits, 3 for more than two visits). The accompanying table shows
joint probabilities.

1
 Number of Years at the university (X)
visits (Y) 1 2 3 4
0 0.06 0.08 0.07 0.02
1 0.08 0.07 0.06 0.01
2 0.05 0.05 0.12 0.02
3 0.03 0.06 0.18 0.04
a) Find and interpret
b) Find and interpret the mean number of X.
c) Find and interpret the mean number of Y.
d) If the randomly selected student is a year student, what is the probability that he or she) visits
museum at least 3 times?
e) If the randomly selected student has 1 visit to a museum, what is the probability that he (or she) is a
year student?
f) Are number of visits to a museum and years at the university independent of each other?

4. It was found that 20% of all people both watched the show regularly and could correctly identify the
advertised product. Also, 27% of all people regularly watched the show and 53% of all people could
correctly identify the advertised product. Define a pair of random variables as follows:
X=1 if regularly watch the show; X=0 otherwise
Y=1 if product correctly identified; Y=0 otherwise.
a) Find the joint probability function of X and Y.
b) Find the conditional probability function of Y, given X=0.
c) If randomly selected person could identify the product correctly, what is the probability that he (or she)
regularly watch the show?
d) Find and interpret the covariance between X and Y.

5. Let the joint probability mass function of random variables X and Y be given by

a) Find the value of constant C,

b) Find the marginal probability functions of X and Y , namely and
c) Find and
d) Are X and Y independent or dependent? Explain your answer

6. A bag has four red and two white balls. Two balls are drawn at random, consecutively,
without replacement, from the bag. Let X be a random variable representing the number of
red balls in the two consecutive draws, and let Y be a random variable representing the
number of white balls in the first draw. The table below is the joint probability distribution
function (pdf) of the two variables X and Y. Determine the values of a,b,and f. (Hint: A
probability tree will help). If you find c,d, and e, you will get bonus points.

2
 0 1 2
0 a b c
1 d e f

7. Find the covariance of the two random variables whose joint pdf shown below:
0 1 2
0 0.1 0.2 0.3
1 0.1 0.1 0.2

8:
Find a and b for the probability distribution function of the random variable X given below, if
E(X) = 2.9
1 2 3 4
a 0.2 0.25 b
9:
a) The random variable X can take on values 3, 4, 5, 6, and 20. The probabilities for these
are the inverses of their values. Find the expected value and variance of X.
b) Let the random variable X represent the number of red balls in two draws without
replacement from a bag of seven balls, three of which are red. What is the pdf of X ? Find
E(X).
c) For the bag in problem (2), if 10 draws are made with replacement, what is the
probability that between two and five red balls will be counted?
(separate questions)
10:
The random variable X can have the values 0 or 1, and the random variable Y can have the
values 0, 1, or 2. The joint probability distribution of X and Y is given below:
0 1 2
0 0.1 A 0.15
1 B 0.1 C

What are A, B, and C if E(X) = 0.45 and E(Y) = 1.00 ?

11. A bag has one white, two blue, and three red balls. Two balls are drawn in succession, at
random, without replacement, from the bag. The random variable X represents the number of blue
balls among the two drawn balls, and the random variable Y represents the number of red balls in
the second draw. The joint probability distribution of X and Y is given below:

0 1 2

3
 1/10 1/15
0 A
3/10 1/5
1 0
a) Find A

b) Find covariance

12.

Two dice are rolled, Let X-be an event “maximum of two numbers obtained, and let Y-be event
minimum of two numbers obtained. Find joint probability distribution table.

13.

A farm contains 7 cows, 10 hens and 8 goats. Thieves has stolen 4 of animals. Make a joint probability
distribution of hens and goats stolen,

14.
Let the joint probability mass function of random variables X and Y be given by

a) Find the value of constant C

b) Find expected value of X and Y , namely and

c) Find covariance between X and Y