Buses arrive at a stop with random times, according to a Poisson Process with rate λ. Mario arrives at the stop at time t. How much time on average $E(W)$ will he has to wait for the next bus' arrive?
(1)The assence of memory of the considered process implies that the distribution of waiting time does not depend on the time of arrival, so $E(W)=1/\lambda$.
(2)The time of arrival is a random time in the interval between two consecutive buses'arrivals. For reasons of symmetry, the time of waiting must be half of the time expected between two consecutive buses, so $E(W)=1/(2\lambda)$.
Choose the right affirmation motivating your answer and expain why the other affirmetion isn't correct.
I think that (1) is correct, because the random variable that describes numbers of arrives has assence of memory. It's correct? If it is, I don't know how show that (2) is wrong. Can somebody help me please?
Thanks.