1. ## markov chain-fair die

A fair die is thrown repeatedly. Let $S_{n}$ be the sum of the outcomes, and let $R_{n}$ be the remindeer when $S_{n}$ is divided by 4(that is $R_{n}$ is the sum of the first n throws reduced modulo 4).
a) Show that $R_{n}$ ia a Markov chain on state space {0,1,2,3}.
b) what are the transitoin probabilities for this chain?
Can you please give me numerical example of $R_{n}$ to see what is going on in this chain as I find it difficult to understand the question.
thanks for any help.

2. Originally Posted by charikaar
A fair die is thrown repeatedly. Let $S_{n}$ be the sum of the outcomes, and let $R_{n}$ be the remindeer when $S_{n}$ is divided by 4(that is $R_{n}$ is the sum of the first n throws reduced modulo 4).
a) Show that $R_{n}$ ia a Markov chain on state space {0,1,2,3}.
b) what are the transitoin probabilities for this chain?
Can you please give me numerical example of $R_{n}$ to see what is going on in this chain as I find it difficult to understand the question.
thanks for any help.
I don't know if this is what you mean by a numerical example but here it goes. Suppose you rolled 4,2,6,1 then R looks like (for n=1,2,3,4) 0,2,0,1.

a) should be relatively simple (use the Markov property of the S_n) and for b), think about it in this way, suppose you are at 0 and want to go to 1. You have to roll either a 1, or a 5 (both give remainder 1). So from 0 to 1 has probability 1/3. Now try to figure the other ones out using the same method.

Hope this helps.