Given cars crossing certain place on a highway follows a Poisson process with rate $\lambda = 3$ mins. David waits to cross the road, but he only crosses if he sees no cars coming by in the next 30 seconds. Find his expected waiting time (hint: condition on the time of the first car).
My attempt: Let $X =$ waiting time of David before he crosses the street (so, $X\geq \frac{1}{2}$ mins), $Y = $ time that the 1st car took to cross the point where David waits (in mins), so $Y$ follows P.P$(3)$
We have: $E(X) = E(X|Y<\frac{1}{2})P(Y<\frac{1}{2}) + E(X|Y\geq \frac{1}{2})P(Y\geq \frac{1}{2})$. The 2nd term is actually just $\frac{1}{2}\ P(N(\frac{1}{2}) = 0) = \frac{1}{2}e^{\frac{-3}{2}},$ because $X$ David would cross the road immediately after waiting for $30$ seconds without seeing any cars.
I'm stucked here because I could not find a way to compute the first term (a more difficult case). Could someone please help me on how to compute this first term?
