This is an irreducible finite markov chain and
P= 0.8 0.2
0.6 0.4
If you solve the limiting distribution you get that in the limit P(sunny)=0.75 and P(rain)=0.25
therefore 0.75n days will be sunny and 0.25n days will be rainy.
Hi everyone, any help in how to work out this problem would be much appreciated...
If it is sunny today, there is a 80% chance it will be sunny tomorrow, and a 20% chance it will be rainy. If it is rainy today, there is a 40% chance it will be rainy tomorrow, and a 60% chance it will be sunny.
In n days, how many of them will be sunny and how many will be rainy?
you first have to get the limiting distributions which you can easily get by calculating P^n for n large enough and taking any row (all rows will be equal)
or solving the matrix [(P-I)`|0] with last row substituted with 1's since probabilities must sum to 1.
in this case:
P-I=
-0.2 0.2
0.6 -0.6
(P-I)`=
-0.2 0.6
0.2 -0.6
solve for:
-0.2 0.6 | 0
1 1 | 1
or -0.2x+0.6y=0 and x+y=1
you will get x=0.75 and y=0.25 if you solve the 2 equations.
these are your limiting probabilities.
let me know if you need more help