On any given day, the weather in a town can be classified as sunny, cloudy or rainy. Assume that there can never be two sunny days in a row and that any sunny day is equally likely to be followed by a cloudy or rainy day, that cloudy and rainy days have a 50% chance of being followed by the same weather on the next day and that if there is change from cloudy or rainy, it is equally likely to be to a sunny day.
(a) Write the probability transition matrix of this Markov chain.
So the matrix I had is:
$$ \begin{matrix} & S & C & R\\ S & 0 & {1\over2} & {1\over2} \\ C & {1\over2} & {1\over2} & 0 \\ R & {1\over2} & 0 & {1\over2} \\ \end{matrix} $$
But my classmates argue that it is:
\begin{matrix} & S & C & R\\ S & 0 & {1\over2} & {1\over2} \\ C & {1\over4} & {1\over2} & {1\over4} \\ R& {1\over4} & {1\over4} & {1\over2} \\ \end{matrix}
So I am guessing I am probably wrong but I don't understand how there is ${1\over4}$ for some of the probabilities. Any help please!
The problem statement says that if there is a change from cloudy or rainy, it is equally likely to be to a sunny day. Your transition matrix doesn’t correspond to this; it corresponds to a change away from cloudy or rainy always leading to sunny weather. Your classmates are right, since in their matrix the two probabilities for changing away from cloudy or rainy weather are the same. Since the overall probability to change away from cloud or rainy weather is $\frac12$, each of the two possible ways of changing away has half that probability, i.e. $\frac14$.