What Is the Probability of Waiting Time T in a Poisson Process?

[Milky]

Homework Statement

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

Homework Equations

The Attempt at a Solution

I know that interarrival times have an exponential distribution.
If X is the arrival of the next car,
P(waiting time = T) = P( X > T )
But I am not sure what this means. Any help would be appreciated.

 

[mighty2000]

Hello there!

First, let's define some terms to make things clearer. The Poisson process is a stochastic process in which events occur independently at a constant rate λ. In this case, the events are cars passing by the street location. The arrival time of each car follows an exponential distribution with the rate parameter λ.

Now, let's look at the question. The woman wants to cross the street at that location and she will wait until she can see that no cars will come by in the next T time units. This means that she will wait until the arrival time of the next car (X) is greater than T. So, the probability that her waiting time is T is equal to the probability that X is greater than T.

We can express this mathematically as follows:

P(waiting time = T) = P(X > T) = e^(-λT)

This is because the probability that X is greater than T is equal to the area under the probability density function of the exponential distribution from T to infinity. And the probability density function of the exponential distribution is given by f(x) = λe^(-λx).

So, the probability that the woman's waiting time is T is equal to e^(-λT). I hope this helps! Let me know if you have any further questions.