Question:Problem 4. Ships

All ships travel at the same speed through a wide canal. Each ship takest days to traverse the length of the canal. Eastbound ships (i.e., ships traveling east) arrive as a Poisson process with an arrival rate ofE ships per day. Westbound ships (i.e., ships traveling west) arrive as an independent Poisson process with an arrival rate ofW ships per day. A pointer at some location in the canal is always pointing in the direction of travel of the most recent ship to pass it.

In each part below, your answers will be algebraic expressions in terms ofE, W, x, t, v and/ork.

For Part 1, suppose that the pointer is currently pointing west.

1. Determine the PDF,fX(x), of the remaining time,X, until the pointer changes direction. Forx0,fX(x)=unanswered Please note that for Part 1, I have already tried the following two answers which are both wrong: E e^(-Et) and e^(-Et) are both wrong answers! For the remaining part of this problem, we make no assumptions about the current direction of the pointer.
2. Starting at an arbitrary time, we monitor the cross-section of the canal at some fixed location along its length. LetVbe the amount of time that will have elapsed (since we began monitoring) by the time we observe our seventh eastbound ship. Find the PDF ofV. Forv0,fV(v)=unanswered Please note that for Part 2, I have already tried the following answer which is wrong: [(E)^7 t^6 e^(-Et)] / 6! This answer is wrong!