Let N(t) be a Poisson process with intensity = 2, and let X 1 , X 2 , be the corresponding interarrival times.

Let N(t) be a Poisson process with intensity λ = 2, and let X₁, X₂, ⋯ be the corresponding interarrival times.

a. Find the probability that the first arrival occurs after t = 0.5, i.e., P(X₁ > 0.5).

b. Given that we have had no arrivals before t = 1, find P(X₁ > 3).

c. Given that the third arrival occurred at time t = 2, find the probability that the fourth arrival occurs after t = 4.

d. I start watching the process at time t = 10. Let T be the time of the first arrival that I see. In other words, T is the first arrival after t = 10. Find ET and Var(T).

e. I start watching the process at time t = 10. Let T be the time of the first arrival that I see. Find the conditional expectation and the conditional variance of T given that I am informed that the last arrival occurred at time t = 9.