©Zarestky
Math 141 Quiz 11
12/3/2009
NAME: _______KEY_________________________________
Use the following information for problems 1−4: At a university, 75% of all first-semester students buy their
books at the Campus Bookstore, and 25% buy their books at the Off-Campus Bookstore. Each semester,
65% of those who bought their books at the Campus Bookstore and 45% of those who bought their books
at the Off-Campus Bookstore in the preceding semester then buy their books at the Campus Bookstore in
the next semester.
1.
Give the transition matrix T and initial state matrix X0 for this situation.
C
O
C ! 0.65 0.45 $
T=
#
&
O #" 0.35 0.55 &%
2.
X0 =
C
O
! 0.75 $
#
&
#" 0.25 &%
What percent of the market will each bookstore have after 1 semester? (Round the percent to 2
decimal places if necessary.)
! 0.65 0.45 $ ! 0.75 $ ! 0.6 $
X 1 = TX 0 = #
&#
&=#
&
#" 0.35 0.55 &% #" 0.25 &% #" 0.4 &%
60% will buy from the Campus bookstore and 40% will buy from the Off-campus bookstore after 1
semester.
3.
What percent of the market will each bookstore have after 4 semesters? (Round the percent to 2
decimal places if necessary.)
4
! 0.65 0.45 $ ! 0.75 $ ! 0.5628 $
X4 = T X0 = #
& #
&=#
&
#" 0.35 0.55 &% #" 0.25 &% #" 0.4372 &%
56% will buy from the Campus bookstore and 44% will buy from the Off-campus bookstore after 4
semesters.
4
4.
What is the steady state distribution?
(
)
Solve TX = X, i.e. T ! I X = 0 , incorporating the condition that the columns sum to 1.
" 0.65 ! 1 0.45 0 %
" 1 0 0.5625 %
$
'
$
'
$ 0.35 0.55 ! 1 0 ' ( rref ( $ 0 1 0.4375 '
$#
$# 0 0
'&
1
1
1 '&
0
56.25% will buy from the Campus bookstore and 43.75% will buy from the Off-campus bookstore in the
long-run.