I'm having some trouble figuring out the Transition Matrix for the following problem:
"Consider a coin flipping game. On each game 2 coins are flipped simultaneously. Admit Xn is the number of tails obtained on the first n games."
We can get 0,1 or 2 tails right? Then the matrix would be:
\begin{matrix} 1/4 & 2/4 & 1/4 \\ 0 & 1/4 & 1/4 \\ 0 & 0 & 1/4 \end{matrix}
But I have a feeling this is incorrect...
Could anyone shed some light on this problem?
Thank you!
In each game we can have zero tails with probability $1/4$, one tail with probability $1/2$, or two tails with probability $1/4$. It follows then that the transition probabilities for $X_n$ are given by $$ P_{ij} = \begin{cases}1/4,& i=j\\ 1/2,& j=i+1\\ 1/4,& j=i+2. \end{cases} $$ This is a transient Markov chain as there are no backward transitions, so not much can be said about limiting behavior (other than that $\lim_{X_n}=+\infty$ with probability one.