Exercise 16.1. Two phone companies, let's call them P, and P₂, are competing for customers in the Bay Area. Suppose that during each calendar year, 85% of the customers of P₁ stay with P1 and 15% of Pi's customers switch to P2, and that during each calendar year, 92% of the customers of P2 stay with P2 and 8% of their customers switch to P₁. (a) For each n ≥ 0, define the vector Vn n, and Yn In = where x,, is the number of customers for P₁ in year , Yn xn+1 = (0.85)x + (0.08)yn, yn+1 = (0.15)xn+ (0.92)yn, is the number of customers of P2 in year n. Explain why - and write a 2 x 2 Markov matrix M for which vn+1 Mv, for all n ≥ 0. I (b) For the correct Markov matrix M that you should have found in (a), direct computation (which 7345 1416] .2655 .8584| we are not asking you to do here) shows that M² What does the first row of this matrix tell us concerning the behavior of customers of each of P₁ and P2 after two years have elapsed? (A customer who began with one of these companies and is with it at the end of two years may have switched to the other company in the middle of this time, but don't try to keep track of that level of detail.) (c) Again using the correct matrix M from (a), one can show that Mm m. Interpret the meaning of the two numbers appearing in this matrix. 1.3478 3478] .6522 .6522 for large
Exercise 16.1. Two phone companies, let's call them P, and P₂, are competing for customers in the Bay Area. Suppose that during each calendar year, 85% of the customers of P₁ stay with P1 and 15% of Pi's customers switch to P2, and that during each calendar year, 92% of the customers of P2 stay with P2 and 8% of their customers switch to P₁. (a) For each n ≥ 0, define the vector Vn n, and Yn In = where x,, is the number of customers for P₁ in year , Yn xn+1 = (0.85)x + (0.08)yn, yn+1 = (0.15)xn+ (0.92)yn, is the number of customers of P2 in year n. Explain why - and write a 2 x 2 Markov matrix M for which vn+1 Mv, for all n ≥ 0. I (b) For the correct Markov matrix M that you should have found in (a), direct computation (which 7345 1416] .2655 .8584| we are not asking you to do here) shows that M² What does the first row of this matrix tell us concerning the behavior of customers of each of P₁ and P2 after two years have elapsed? (A customer who began with one of these companies and is with it at the end of two years may have switched to the other company in the middle of this time, but don't try to keep track of that level of detail.) (c) Again using the correct matrix M from (a), one can show that Mm m. Interpret the meaning of the two numbers appearing in this matrix. 1.3478 3478] .6522 .6522 for large
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Exercise 16.1. Two phone companies, let's call them P, and P₂, are competing for customers in the Bay
Area. Suppose that during each calendar year, 85% of the customers of P₁ stay with P1 and 15% of Pi's
customers switch to P2, and that during each calendar year, 92% of the customers of P2 stay with P2 and
8% of their customers switch to P₁.
(a) For each n ≥ 0, define the vector Vn
n, and
Yn
In
=
where x,, is the number of customers for P₁ in year
,
Yn
xn+1 = (0.85)x + (0.08)yn, yn+1 = (0.15)xn+ (0.92)yn,
is the number of customers of P2 in year n. Explain why
-
and write a 2 x 2 Markov matrix M for which vn+1 Mv, for all n ≥ 0.
I
(b) For the correct Markov matrix M that you should have found in (a), direct computation (which
7345 1416]
.2655 .8584|
we are not asking you to do here) shows that M²
What does the first row of
this matrix tell us concerning the behavior of customers of each of P₁ and P2 after two years have
elapsed? (A customer who began with one of these companies and is with it at the end of two
years may have switched to the other company in the middle of this time, but don't try to keep
track of that level of detail.)
(c) Again using the correct matrix M from (a), one can show that Mm
m. Interpret the meaning of the two numbers appearing in this matrix.
1.3478 3478]
.6522 .6522
for large](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F425fa4f2-b6ef-4aa0-861e-803b807cec46%2F43ef1fe5-02a2-4e2f-aa02-b6169aebba6d%2Fqmwkrn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 16.1. Two phone companies, let's call them P, and P₂, are competing for customers in the Bay
Area. Suppose that during each calendar year, 85% of the customers of P₁ stay with P1 and 15% of Pi's
customers switch to P2, and that during each calendar year, 92% of the customers of P2 stay with P2 and
8% of their customers switch to P₁.
(a) For each n ≥ 0, define the vector Vn
n, and
Yn
In
=
where x,, is the number of customers for P₁ in year
,
Yn
xn+1 = (0.85)x + (0.08)yn, yn+1 = (0.15)xn+ (0.92)yn,
is the number of customers of P2 in year n. Explain why
-
and write a 2 x 2 Markov matrix M for which vn+1 Mv, for all n ≥ 0.
I
(b) For the correct Markov matrix M that you should have found in (a), direct computation (which
7345 1416]
.2655 .8584|
we are not asking you to do here) shows that M²
What does the first row of
this matrix tell us concerning the behavior of customers of each of P₁ and P2 after two years have
elapsed? (A customer who began with one of these companies and is with it at the end of two
years may have switched to the other company in the middle of this time, but don't try to keep
track of that level of detail.)
(c) Again using the correct matrix M from (a), one can show that Mm
m. Interpret the meaning of the two numbers appearing in this matrix.
1.3478 3478]
.6522 .6522
for large
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