Discussion
Suppose \(\{X(t), t \ge 0\}\) is a pure birth process with birth rate \(\lambda_{i} = \lambda\) for all \(i \in S\) and \(X(0) = 0\). The following are four statements.
(I) \(\{X(t), t \ge 0\}\) is a process with stationary and independent increments.
(II) \(E(X(t)) = e^{\lambda t}\)
(III) X(t) follows Poisson P(\(\lambda t\)) distribution for every fixed t.
(IV) X(0) cannot be 0 in a pure birth process.
Which of the following is true ?
Answer: C
Only (I) and (III) are true.
A pure birth process with constant rate \(\lambda\) starting at 0 is the Poisson process. It has stationary and independent increments (I), and X(t) is Poisson with mean \(\lambda t\) for each t (III).
(II) is false: the mean is \(E(X(t)) = \lambda t\), not \(e^{\lambda t}\). The exponential mean belongs to the Yule process, where the birth rate is proportional to the population, \(\lambda_{i} = i\lambda\). (IV) is false: X(0) = 0 is the usual starting point of the Poisson process.
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