I am studying stochastic process written by Sheldon M Ross.
I have a question about the conditional distribution of the arrival time.
Theorem 2.3.1
$$S_1, S_2, \cdots, S_n |_{N(t)=n} \sim U_1, U_2, \cdots, U_n$$ where \begin{align} S_i&\mbox{: i-th arrival time, } ~i=1, 2, \cdots, n \\ U_i&\mbox{: ordered Uniform distribution in time } t\in (0, t), ~i=1, 2, \cdots, n \\ \end{align}
The textbook proved using the fact that joint probability of $S_1, S_2, \cdots, S_n |_{N(t)=n}$ is same as that of $U_1, U_2, \cdots, U_n$ like:
\begin{align} f_{U_1, U_2, \cdots, U_n}(u_1, u_2, \cdots, u_n)=\frac{n!}{t^n}\\ f_{S_1, S_2, \cdots, S_n|N(t)=n}(s_1, s_2, \cdots, s_n|n)=\frac{n!}{t^n}\\ \end{align}
I understand the above. However, I am wondering whether $$S_i |_{N(t)=n} \sim U_i$$ is correct or not.
Thank you for reading my question, and I am sorry about whenever I am asking.... without answering.