conditional expectation value of poisson process

Question

I consider a poisson process with rate $$\lambda$$ . Since the Inter arrival time follows exponential distribution, the expectation value of the inter arrival time, say $$E[X]= \frac{1}{\lambda}$$. If I look particularly for a time interval [0-15s]. Then,

1. What would be the expectation value of the inter arrival time e.g., first, second, third arrivals,......? Would it be $$E[X| t<15]$$

2. What is the expectation value of the inter arrival time if I can it is conditioned over [5s-15s] i.e., $$E[X| 5<t<15]$$?

Answer 1

Are you looking for the expected inter arrival time when given that that the next event occurs within a particular interval?

$$\begin{align*}\mathsf E(X\mid s\leqslant X\lt t) ~&=~ \dfrac{\int_s^t \lambda x \mathrm e^{-\lambda x}\operatorname d x}{\int_s^t \lambda \mathrm e^{-\lambda x}\operatorname d x} \\[2ex] &=~ \dfrac{(\lambda s+1) \mathrm e^{-\lambda s}-(\lambda t+1)\mathrm e^{-\lambda t}}{\lambda (\mathrm e^{-\lambda s}-\mathrm e^{-\lambda t})}\end{align*}$$