Poisson Process With Conditioning?

Question

Cars pass a certain street location according to a Poisson process with rate Lamda. A woman who wants to cross the street at that location waits until she can see that no cars will come by in the next T time units.
A) Find the probability that her waiting time is 0
B) Find the expected waiting time

Hint: Condition on the time of the first car

Answer 1

(A) The probability that a wait will be necessary is the same as the probability that a car comes within the next T seconds: 1 - exp(-lambda T). So the probability of no wait is:

P = exp(-lambda*T)

(B) Let Q be the expected waiting time. If t is the time until the next car arrives (t < T), the expected additional wait time will be Q, giving an expected total wait time of t + Q. In other words the expectation value of the total wait time is:

Q = \integral_0^T (lambda exp(-lambda*t))(t + Q) dt
= [-exp(-lambda t)(Q + 1/lambda + t]_0^T
= Q + 1/lambda - exp(-lambda T)(Q + 1/lambda + T)

Solving for Q gives the answer:

Q = (exp(+lambda T) - 1)/lambda - T

Answer 2

GAACHOOON!