Question

1. A store stocks a particular item. The demand for the product each day is 1 item with probability 1/6th, 2 items with probability 3/6th, and 3 items with probability 2/6th. Assume that the daily demands are independent and identically distributed. Each evening if the remaining stock is less than 3 items, the store orders enough to bring the total stock up to 6 items. These items reach the store before the beginning of the following day. Assume that any demand is lost when the item is out of stock.

   1. (a) Let Xn be the amount in stock at the beginning of day n; assume that X0 = 5.

   2. (b) Let Yn be the amount in stock at the end of day n; assume that Y0 = 2.

2. For the above problem, compute the following:

   (a) Pr{X2 =6 | X0 =5}.
   (b) Pr{X2 =5,X3 =4,X5 =6 | X0 =3}.

   (c) E(X2 | X0 = 6).
   (d) Assume the initial distribution is (0, 0, .5, .5). Find Pr {X2 = 6}.

   (e) With the same initial distribution in 2d, find Pr {X4 = 3, X1 = 5 | X2 = 6}

Answer 1

Solution:

1)

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