Problem Statement
A subscription service classifies subscribers as either Active ($A$) or Inactive ($I$). In any given month, $85\%$ of active subscribers remain active while $15\%$ become inactive. Of the inactive subscribers, $20\%$ reactivate their subscription while $80\%$ remain inactive.
Initially, $70\%$ of registered users are Active and $30\%$ are Inactive.
**(a)** [2 marks] Write down the transition matrix $\mathbf{T}$ and the initial state probability vector $\mathbf{s}_0$.
**(b)** [2 marks] Find the proportion of active subscribers after 3 months.
**(c)** [3 marks] Determine the long-term steady-state proportion of active subscribers.
Verified Solution & Marking Scheme
Part (a) [A1]
$\mathbf{T} = \begin{pmatrix} 0.85 & 0.15 \\ 0.20 & 0.80 \end{pmatrix}$
Part (a) [A1]
$\mathbf{s}_0 = \begin{pmatrix} 0.70 & 0.30 \end{pmatrix}$
Part (b) [M1]
$\mathbf{s}_3 = \mathbf{s}_0 \mathbf{T}^3 = \begin{pmatrix} 0.70 & 0.30 \end{pmatrix} \begin{pmatrix} 0.85 & 0.15 \\ 0.20 & 0.80 \end{pmatrix}^3$
Part (b) [A1]
$\mathbf{s}_3 \approx \begin{pmatrix} 0.587 & 0.413 \end{pmatrix} \implies 58.7\%$ active
Part (c) [M1]
Set $\begin{pmatrix} p & 1-p \end{pmatrix} \begin{pmatrix} 0.85 & 0.15 \\ 0.20 & 0.80 \end{pmatrix} = \begin{pmatrix} p & 1-p \end{pmatrix}$
Part (c) [M1]
$0.85p + 0.20(1 - p) = p \implies 0.85p + 0.20 - 0.20p = p \implies 0.20 = 0.35p$
Part (c) [A1]
$p = \frac{0.20}{0.35} = \frac{4}{7} \approx 57.1\%$ long-term active subscribers