You are the fresh produce manager for a large supermarket. Demand for a fresh lettuce for next week is uncertain and the uncertainty can best be described by the following pdf for X: fx (x)=(0,e where X = demand for lettuce (measured in thousand pounds) for next week.
As a manager you need to decide how much lettuce should you stock. If you stock too much, there will be some unsold lettuce that is wasted (lettuce is perishable and won't store) and if you stock too little there will be unmet demand resulting in lost profits. Suppose you decide to stock q units of lettuce, where q> 0.
a) What is the expected demand for lettuce, i.e., compute E(X).
b) Find k such that p(X ≤k) = p(X2k). Is k the same as E(X)?
c) What is the probability that the initial stock q will not be enough to meet the demand?
d) What is the expected value of the stock left over at the end of the week?
e) What is the expected quantity sold for next week?
Note: Quantity sold is different from demand (X). Quantity sold is equal to X if there is enough stock. If the manager selects a smaller q, then quantity sold may be capped at q even though demand exceed q.
f) How do you decide on optimal q? This is an open-ended question; I am not looking for any particular "correct" answer. Any reasonable way to select q is accepted.