Poisson Process: interevent times

[SantyClause]

Homework Statement

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 (ex/u -1)/lambda


I know we need x/u seconds to cross, but I really don't even know how to start it :/ Will we use the erlang distribution?

 

[Ray Vickson]

Homework Statement

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 (ex/u -1)/lambda


I know we need x/u seconds to cross, but I really don't even know how to start it :/ Will we use the erlang distribution?

What do you mean by the notation (ex/u -1)/lambda? Do you mean $(e (x/u) - 1)/\lambda$, or $(e^{x/u}-1)/\lambda$ or $e^{(x/u) - 1}/\lambda$, or something else?

Have you heard of the "memoryless" property of exponential distributions? You need to use it.

As to how to start: let T be the amount of time that passes until she starts crossing the road. Under what conditions is T = 0? If T > 0, she will wait for the next car to pass and then start again.