A survey of retirees in Arizona who golf or fish everyday found that 70% of thos
ID: 3037319 • Letter: A
Question
A survey of retirees in Arizona who golf or fish everyday found that 70% of those that played golf one day switched to fishing the next day and that 80% of those who went fishing one day switched to golf the next day. Twenty percent of retirees are golfing today.
a) Rewrite the given data into an initial distribution matrix and a transition matrix. Label each matrix appropriately (rows, columns, and the matrices themselves)
b) If 20% of retirees are golfing today, what percentage of retirees will be golfing the day after tomorrow?
c) What proportion of retirees will be golfing on a given day in the long run?
PLEASE TYPE SOLUTION
Explanation / Answer
(a). The initial distribution matrix is
G
F
G
30%
80%
F
70%
20%
where G denotes golfing and F denotes fishing.
The transition matrix is M =
G
F
G
0.30
0.80
F
0.70
0.20
(b).Let the population of retirees in Arizona be x. Then 0.2x people are playing golf today and 0.8xz are fishing today. If X = (0.2x,0.8x)T, then the number of retirees who are golfing and fishing, respectively, tomorrow is given by MX = (0.70x, 0.30x) and the number of retirees who are golfing and fishing, respectively, day after tomorrow is given by MMX = (0.45x, 0.55x)T which means that 45% of retirees will be golfing day after tomorrow.
(c ). The steady state vector X is given by the equation MX = X or, (M-I2)X = 0. The matrix M-I2 is
-0.70
0.80
0.70
-0.80
The RREF of this matrix is
1
-8/7
0
0
Now, if X = (x,y)T, then the equation (M-I2)X = 0 is equivalent to x -8y/7 = 0 or, x = 8y/7. Then X = (8y/7,y)T = (y/7)(8,7)T. Thus, the steady state vector is (8,7)T. This means that, in the long run, on a given day, 8/15th of the retirees will be golfing.
G
F
G
30%
80%
F
70%
20%
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