Car Convoy Problem Cars travel down a one-way single-track road. The nth driver would like to drive at speed $V_n$ , where $ V_1,V_2, ..., V_n$ are iid random variables. Cars will get bunched into convoys. If $V_2>V_1>V_3$ , then the first convoy will consists of cars 1 and 2 , and will be of length 2. Let L be the length of the first convoy. Find the probability $P(L=n)$ and the expectation $E(L)$.

Hint:

* For the first convoy to be of length $L=k$, you need cars $2,3,\ldots,k$ to all be faster than car $1$, and car $k+1$ to be slower than car $1$
* As special cases, for the first convoy to be of length $L=1$ you need car $2$ to be slower than car $1$, and for the first convoy to be of length $L=n$ you need all the other cars $2,3,\ldots,n$ to be faster than car $1$