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
xt - 0,6*xt = 0,2
0,4*xt = 0,2
xt = 0,5 and xt+1 = 0,5
b) If in January 0,5 ( 50%) of customesr were low user
In february teh % of low user customer wll be
xt = 0,8* xt + 0,2* xt+1
0,5 = 0,8*0,5 + 0,2*xt+1
0,5 = 0,4 + 0,2 *xt+1
0,1/0,2 = xt+1 xt+1 = 0,5 or 50%
The same answer for March
will be a polynomial