Discussion
Let \(\{X(t), t \ge 0\}\) be a time homogeneous Poisson process with rate \(\lambda\). Let \(X(T) = n \gt 0\) and let \(T_{1} \lt \cdots \lt T_{n} \lt T\) be times at which these \(n\) events occur. Then, \(E(T_{1} \mid X(T) = n)\) is given by :
Answer: A
\(E(T_{1} \mid X(T) = n) = \) \(\dfrac{T}{n + 1}\).
For a (homogeneous) Poisson process, given that \(n\) events occurred in [0, T], the event times are distributed as the order statistics of \(n\) independent \(U(0, T)\) variables. The smallest of \(n\) uniforms on [0, T] has mean
\[ E(U_{(1)}) = \frac{T}{n + 1} \]
(the \(n\) points split [0, T] into \(n + 1\) gaps of equal expected length). The answer does not depend on \(\lambda\), which rules out \(T\lambda^{n}\), and \(n/(n + 1) - T\) can be negative.
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