When Linda walks from home to work, she has to cross the street at a certain point. Linda needs a gap of 15 seconds in the traffic to cross the street at that point. Suppose that the traffic flow is a...

When Linda walks from home to work, she has to cross the street at a certain point. Linda needs a gap of 15 seconds in the traffic to cross the street at that point. Suppose that the traffic flow is a Poisson process, and the mean time between two consecutive cars passing by Linda is 7 seconds. Find the expected value of the time Linda has to wait before she can cross the street.

Hint: Let X₁
be the time between Linda’s arrival at the point and the first car passing by her. Let X₂
be the time between the first and second cars passing Linda, and so forth. Let N be the first integer for whi󀁨

X₁
≤ 15, X₂
≤ 15, . . . , X_N
≤ 15, X_N+1
> 15.

The time Linda has to wait before being able to cross the street is 0 if N = 0 (i.e., X₁
> 15) and is S_N
= X₁
+ X₂
+ · · · + X_N
otherwise. Calculate SN by conditioning on N.