NAME:
SOLUTION to Midterm II_
1 of 4 pages
5/25/2007
STAT 427
Midterm II
Spring, 2007
Prof. Goel
MWF 9:30 - 10.20
Problem 1.
[10 points]
A gas station has two islands: Island #1 has six pumps, and Island #2 has
four pumps. Let the random variables X and Y denote:
X = number of pumps in use at Island #1, and Y = number of pumps in use at Island #2.
a.
[2 points]
Describe the set of all possible values of
(
x,y
)
.
{
(
, )
:
{0,1,2,3,4,5,6},
{0,1,2,3,4}}
x
y
xy
∈∈
A total of 7x5 = 35 possible pairs of (
x,y
) values
b.
[2 points]
Give the possible values for
T
= the total number of pumps in use.
T
:
Possible values:
{0,1,2,3,4,5,6,7,8,9,10}
c.
[2 points]
Give the possible values for A
D
= the absolute difference between the
numbers of pumps in use at Islands 1 and 2.
AD: Possible values: {0,1,2,3,4,5,6}
d.
[2 points]
Z
= the number of islands having exactly two pumps in use.
Z
: Possible values:
{0,1,2}
e.
[2 points]
Explain why X and Y may not be independent random variables.
It is unlikely that
P
[
Y
=4|
X
=0] is equal to
P
[
Y
=4].
Similarly it is unlikely that
P
[
X
=6|
Y
=0] is equal to
P
[
X
=6].
Hence, the two random variables may not be independently distributed.
Problem 2. [20 Points]
(i) [8
Points] Twenty-five percent of all telephones of a certain brand are submitted for
service while under warranty.
Of these, 60% can be repaired whereas the other
40% must be replaced with new units.
If a company purchases ten of these