Let $\{N_{t}, t \geq 0\}$ be a Poisson process of rate $\lambda=5$ and $\{T_{t}, n \geq 1\}$ its corresponding arrival times. It is given the following situations:
- $E(N_{10}| N_{1}=1, N_{3}=4)$ and
- $P(T_{2}=3, T_{4}=4.5, T_{5} = 5)$.
How can I calculate each of them? I am not sure of the exact steps to achieve the expected results.