(i) Suppose that buses arrive at a bus-stop as a Poisson process \{X_t\}_{t\geq 0} of parameter \lambda per hour, and that after an hour exactly \(n\) buses have arrived. Calculate the conditional probabilities \(P(X_t = k | X_1 = n)\) that exactly \(k\) buses, \(0 \leq k \leq n\), have arrived at the bus-stop by time \(t\), \(0 \leq t \leq 1\), given that \(n\) have arrived by time 1.
(ii) Consider a continuous time Markov chain \{X_t\}_{t\geq 0} with two states, 0 and 1. For each \(t \geq 0\) and \(i, j \in \{0, 1\}\), let \(p_{ij}(t) = P(X_t = j | X_0 = i)\). Suppose that for some \(T > 0\) the matrix \(P(T)\) has the form
\(P(T) = \begin{pmatrix} \alpha & 1 - \alpha \\ 1 - \alpha & \alpha \end{pmatrix}\).
Prove that \(1/2 < \alpha \leq 1\), and compute \(P(t)\) for all \(t \geq 0\).
