Inspection Paradox in Poisson Renewal Processes

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Renewal Theory / Inspection Paradox.

Your Task
Buses arrive at a bus stop according to a Poisson process with an average rate of 1 bus every 10 minutes ($\lambda = 0.1 \text{ min}^{-1}$). A passenger arrives at the bus stop at a randomly chosen time $t$ (where $t$ is large).
(1) What is the expected time the passenger must wait until the next bus arrives?
(2) What is the expected length of the total interval between the bus immediately before and the bus immediately after the passenger's arrival? Explain the apparent paradox.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.