more typical bell-shaped distributions. The distributions for suggest that meetings are likely to continue into the next hour but are unlikely to start the following hour. The distributions for $\lambda=0.85$ are shaped similarly to those for $\lambda=1.0$ but are markedly subdued with expected values closer to $0.5$.

![\[fig:Transition-probabilities\]Probability densities for the occupied-to-occupied (blue) and vacant-to-occupied (red) at (a) and (b). Two cases are shown: a heavy forgetting factor ($\lambda=0.85$, solid) and no forgetting ($\lambda=1.0$, dashed). ](nightVsDayDistr2)

Transition Matrix and Occupancy Prediction
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Recall from Section \[sub:Markov-chain-formulation\] and Figure \[fig:markovChain\]b the Markov chain has states $1_{0}\ldots1_{23}$ and $0_{0}\ldots0_{23}$. The probability distribution of the current occupancy state $\pi_{k}\in\mathbb{R}^{1\times2M}$ evolves according to $\pi_{k+1}=\pi_{k}P$. The matrix $P$ can be constructed from the four blocks: $P^{(\textrm{\Romannum{1})}}$ for $1_{k}\to1_{k+1}$, $P^{(\textrm{\Romannum{2}})}$ for $1_{k}\to0_{k+1}$, $P^{(\textrm{\Romannum{3}})}$ for $0_{k}\to1_{k+1}$, and $P^{(\textrm{\Romannum{4}})}$ for $0_{k}\to0_{k+1}$ transitions. The entries for $P^{(\textrm{\Romannum{1}})}$ and $P^{(\textrm{\Romannum{4}})}$ are the expected values of $p_{0\ldots M-1}$ and $q_{0\ldots M-1}$, and the other two matrices are their complements. $$\begin{aligned}
P_{ik}^{\textrm{(\Romannum{1})}} &=
  \begin{cases}
    \makebox[\foo][l]{${\mathbb{P}\left(\gamma_k=1\mid\gamma_i=1\right)}={\mathbb{E}\left[p_i\right]}$} & (\dagger) \\
    0 & \textrm{otherwise}
  \end{cases}\\P_{ik}^{\textrm{(\Romannum{2})}} &= 
\begin{cases}
    \makebox[\foo][l]{${\mathbb{P}\left(\gamma_k=0\mid\gamma_i=1\right)}=1-{\mathbb{E}\left[p_i\right]}$} & (\dagger) \\
    0 & \textrm{otherwise}
  \end{cases}\\P_{ik}^{\textrm{(\Romannum{3})}} &= 
\begin{cases}
    \makebox[\foo][l]{${\mathbb{P}\left(\gamma_k=1\mid\gamma_i=0\right)}={\mathbb{E}\left[q_i\right]}$} & (\dagger) \\
    0 & \textrm{otherwise}
  \end{cases}\\P_{ik}^{\textrm{(\Romannum{4})}} &= 
\begin{cases}
    \makebox[\foo][l]{${\mathbb{P}\left(\gamma_k=0\mid\gamma_i=0\right)}=1-{\mathbb{E}\left[q_i\right]}$} & (\dagger) \\
    0 & \textrm{otherwise}
  \end{cases}\end{aligned}$$where 