# Thread: Markov chain -state space and transition matrix

1. ## Markov chain -state space and transition matrix

A beer vendor adopts the following stock replenishment policy: every Monday morning he
buys fixed number N of six-packs of beer from the distributor, and sells the beer during the
week. No matter how many packs he sells during that week, next Monday morning he will
buy again N six-packs. Assume that the demand Dn for beer on week n is independent of the demand in previous weeks, and that its distribution is given by P(Dn=i) = di, i = 0, 1, 2, . . . (i is the number of six-packs).
Let Xn be the number of beer packs in the store at the end of the nth week.
Show that
{Xn} is Markov chain, and determine its state-space and transition matrix.

2. ## Hint

I have a starting hint that I think should be used for the problem:

Xn can be defined as:
Xn=Xn-1(starting inventory for week n)+N(no of 6 packs of beer purchased at the start of nth week)-Dn(Demand during week n)

How do I go about finding the transition matrix for Xn and state space from here?