A cell phone provider classifies its customers as Low users (less than 400 minutesper month) or High users (400 or more minutes

Question

3 years ago

15

A cell phone provider classifies its customers as Low users (less than 400 minutesper month) or High users (400 or more minutes

per month). Studies have shownthat 80% of the people who were Low users one month will be Low users the nextmonth, and that 70% of the people who were High users one month will be Highusers the next month.
(a) Set up the 2x2 stochastic matrix with columns and rows labeled L and H that displays these transitions
(b) Suppose that, during the month of January, 50% of the customers were Low users. What percent of the customers will be Low users in February? In March? (Type an integer or decimal for each matrix element.)

Mathematics

1 answer:

Svet_ta [14] · Answer 1 · 2022-05-25T01:16:33+03:00

Answer:

a ) See step by step explanation

b) 50% and 50%

Step-by-step explanation:

The definition of type of user x is:

if x < 400 minutes per month low user

if x > 400 minutes per month high user

The value for these states are of course yes or no ( the user is low or high user)

Then:

if the user is low today xt (0) we have the probability of 80% ( 0,8) of have the user in the same condition and 20% (0,2) of having him or her as high user

xt = 1

If the user is high today xt (1) is high today we have the probability of 70% ( 0,7) of having the user in the same condition and 30% (0,3) of having him or her as high user

a) Then we construct the stochastic matrix

Type of user

(xt) xt xt+1

Low (0) or L 0,8 0,2

High(1) or H 0,3 0,7

The resulting equation system is:

xt = 0,8* xt + 0,2* xt+1

xt+1 = 0,3*xt + 0,7* xt+1

1 = xt + xt+1

Solving for variables xt+1 = 1 - xt

xt = 0,8*xt + ( 1 - xt ) * 0,2

xt - 0,8*xt = 0,2 - 0,2xt