poisson cars

[radosr]

If a car passes at the crosswalk on average every 10 seconds and you need 20 seconds to pass the road, how long does it take you on average to cross the road?

 

[Brad Warren]

The process is Markov, so we can condition on the next car passing. The time to cross the road is 20 seconds if no cars pass for 20 seconds. Otherwise, the expected time to cross is the time the car passes plus the original expectation. The time a car passes is distributed as Exp(lambda = 1/10).

(E[T]=\int_0^{20}{(t+E[T])dP(\tau=t)}+\int_{20}^{\infty}{20dP(\tau=t)})
(E[T]=\int_0^{20}{t\lambda e^{-\lambda t}dt}+E[T]\int_0^{20}{dP(\tau=t)}+20\int_{20}^{\infty}{dP(\tau=t)})
(E[T]=10-20e^{-2}-10e^{-2}+E[T](1-e^{-2})+20e^{-2})
(E[T]=10(e^2-1)=63.89)