• A transition probability matrix is said to be doubly stochastic if each row

and column sum up to one (1). Let P be an n n-doubly stochastic transition

probability matrix. Show that P

2 = P P is also doubly stochastic transition. [5 Marks]

• Suppose a shop has a policy that when its stock for a given product is at s or below

it, it will be restocked to S. This restock is assumed to take place early at the

beginning of a day. Let Xn denote the amount of stock at hand at the end of period

n and Dn be the demand on day n. Assume the D = (Dn)n1 is independent and

identically distributed according to a Poisson distribution with = 1.

• Write down an expression for Xn in terms of Dn, Xnn1, S and s. [3 Marks]
• Write down the transition probability matrix for the chain with

s = 1 and S = 3. [5 Marks]

• Show that the chain is irreducible. [2 Marks]
• Find the proportion of time that stock level stood at 1.