10. A recently built bridge was carefully structured to survive the strengths of the most probable earthquakes. Suppose that (1) the bridge will be maintained properly, and no catastrophe other than earthquake would ruin it; (2) for a small p > 0, the probability is p that an earthquake occurring in the seismic region of the bridge is so intensive that the bridge cannot withstand it; and (3) the time until the next earthquake and the times between consecutive earthquakes are independent exponential random variables each with parameter λ. Determine the distribution of the lifetime of the bridge.

Hint: Let X1 be the time until the first earthquake. For i ≥ 2, let Xi be the time between the (i−1)st and the ith earthquakes. Let N be the number of earthquakes until the one that the bridge cannot withstand. The lifetime of the bridge is L =

PN k=1 Xk.

To identify the distribution function of L, calculate its moment-generating function