Problem 3: The Village Well {20 points) A village sources their water from a common well with height h. Every night, the well naturally lls (the well only lls at night}, and every day, villagers collect water from the well (only during the day). Given some current water level in the well: assume that the amount that it lls overnight (Y) and the amount remaining at the end of the day X are both uniformly distributed. Initially, the well is empty. Starting with nighttime, consider the process for the rst 24-hour cycle. (Note: leave your solutions in terms of h) (i) Find the joint PDF of Y 85 X for day 1 (ii) Find the marginal PDF of X (iii) Find the expected water level at the end of the day (iv) (Bonus +2) Evaluate E[X], by exploiting the relation X = Y x: (X /Y) (v) (Bonus +1)What will the eventual water level in the well be after enough days