Each day, Quintana watches exactly one movie. She only watches action movies (genre 1), comedies (genre 2), and dramas (genre 3). Let Xᵢ, i=1,… be a sequence of random variables with Xᵢ=n if Quintana watches a movie of genre n on day i. Suppose that {Xᵢ} is a Markov process with probability transition matrix $$\left( \begin{array} { c c c }
.2 & .27 & .53 \
.18 & .33 & .59 \
.22 & .1 & .67
\end{array} \right)$$ Find the long run proportion of days she watches each genre.

Question

Each day, Quintana watches exactly one movie. She only watches action movies (genre 1), comedies (genre 2), and dramas (genre 3). Let Xᵢ, i=1,… be a sequence of random variables with Xᵢ=n if Quintana watches a movie of genre n on day i. Suppose that {Xᵢ} is a Markov process with probability transition matrix $$\left( \begin{array} { c c c } 
.2 & .27 & .53 \
.18 & .33 & .59 \
.22 & .1 & .67
\end{array} \right)$$ Find the long run proportion of days she watches each genre.

Quizplus · Accepted Answer

The Answer of Each day, Quintana watches exactly one movie. She only watches action movies (genre 1), comedies (genre 2), and dramas (genre 3). Let Xᵢ, i=1,… be a sequence of random variables with Xᵢ=n if Quintana watches a movie of genre n on day i. Suppose that {Xᵢ} is a Markov process with probability transition matrix $$\left( \begin{array} { c c c } 
.2 & .27 & .53 \
.18 & .33 & .59 \
.22 & .1 & .67
\end{array} \right)$$ Find the long run proportion of days she watches each genre.

Each Day, Quintana Watches Exactly One Movie (.2.27.53.18.33.59.22.1.67)\left( \begin{array} { c c c } .2 & .27 & .53 \\.18 & .33 & .59 \\.22 & .1 & .67\end{array} \right)​.2.18.22​.27.33.1​.53.59.67​​

Each Day, Quintana Watches Exactly One Movie $\left( \begin{array} { c c c } .2 & .27 & .53 \\.18 & .33 & .59 \\.22 & .1 & .67\end{array} \right)$