Poisson Process, expected time

Consider a one-way road where the cars form a PP(lambda) with rate lambda cars/sec. The road is x feet wide. A pedestrian, who walks at a speed of u feet/sec, will cross the road if and only if she is certain that no cars will cross the pedestrian crossing while she is on it. Shwo that the expected time until she completes the crossing is (e^{x/u}-1)/lambda

All I know is that it will take x/u seconds to cross. Not sure how to start the problem...

Re: Poisson Process, expected time

Hey SantyClause.

Given this information, you will need to find the probability that corresponds to no cars and then consider the waiting time from that.

The Poisson distribution for P(X = 0) is given by e^(-lambda) which means the number of events you would have to wait would be 1/e^(-lambda) or e^(lambda).

But now you have to factor in the crossing rate with respect to the waiting time for the next event.

The distribution of waiting times is exponential in distribution, so for the waiting time you need P(Waiting Time > Crossing Time) and Crossing Time = Crossing Distance / Crossing Velocity or x/u.

The probability of waiting time P(Waiting Time > x/u) = 1 - P(Waiting Time < x/u) = 1 - (1 - e^(-x/u)) = e^(-x/u)

Now we have to combine both of these pieces of information.

The person can only cross when both of these criteria are met: they have to have no cars in the interval and they also have to have a waiting time that is greater than x/u seconds to the next event.

The final hint I will give is to construct a random variable that gives the time for crossing. You will have two events: one is where you can cross (i.e. the waiting time is > x/u and there are no cars on that interval) and one where you can not.

You can use a compound Poisson process or do something similar by using a form of an indicator variable.