$N$ is a Poisson process with intensity $\lambda$. We have arrival times $T_{1}, T_{2}, \ldots, T_{N},$ independent uniform distributed events in [0,1]. Let $T=\min \left\{T_{1}, T_{2}, \ldots, T_{N}\right\}$. I'm struggling with the following questions:
1- $P(T>t \mid N=n)$ for every $t \in[0,1]$.
2- $E[T \mid N=n]=\frac{1}{n+1}$ , $E[T \mid N]$
3- $E[T]$
I didn't find much examples about the conditional expectation with Poisson processes and continuous uniform distribution.