Question

Can you help me with the following MARKOV CHAINS problem, explaining the solution step by step
A sports magazine obtained the following information from its subscribers: during the 1st year as subscribers, 20% cancel their subscription. Of those who have subscribed for one year, 10% cancel their subscription during the second year. Of those who have subscribed for more than 2 years, 4% cancel their subscription in any given year.
a. Define states and transition matrix
b. On average, how long does a person subscribe to this magazine?

Answer 1

a)

The states of the markov chain are:(subscribed,cancelled)=(0,1),say denote these states respectively.

Probability of cancelling=$P[\{cancelled\}\cap \{subscribed \hspace{0.1cm} \hspace{0.1cm} till \hspace{0.1cm} year \hspace{0.1cm}1\}]+P[\{cancelled\}\cap \{subscribed \hspace{0.1cm} \hspace{0.1cm} till \hspace{0.1cm} year \hspace{0.1cm}2\}]+P[Cancelled\hspace{0.1cm} after\hspace{0.1cm} year \hspace{0.1cm} 2]=0.2+0.1*0.8+0.8*0.9*0.04= 0.3088$

Transition matrix:

$\begin{bmatrix} 0.6912 & 0.3088\\ 0&1 \end{bmatrix}$

b)

Let $\pi_{i}$ be the stationary probabilities,i=0,1

Consider the equation:

$0.6912\pi_{0}+0.3088\pi_{1}=\pi_{0}...(*) \newline and \hspace{0.1cm}\pi_{0}+\pi_{1}=1...(**) \newline (*),(**)=>0.6912\pi_{0}+0.3088(1-\pi_{0})=\pi_{0}=>0.6176\pi_{0}=0.3088\newline \pi_{0}=0.5$

Mean time spent being subscribed is $\frac{1}{0.5}=2$ years