Consider three random variables X, Y, Z, each of which is not lattice and has finite expectation. A process starts in state A. After a random time X, it moves to state B. Then after a random time Y, it moves to state C, and after a random time Z it moves back to A.This process repeats, with the random times chosen a new from the distributions of X, Y and Z each time. In the long run, what fraction of time is spent in state A?
In this question, every time distribution is changes. Can anyone explain me the direction of this question.
Let $\mu_x$, $\mu_y$, and $\mu_z$ be the means of the three time distributions of $X,Y$, and $Z$.
As the "formulas" for the proportion of time spent in any of the three states are symmetric, it comes intuitively that the proportion of time spent in state $A$ is $\displaystyle \frac{\mu_x}{\mu_x+\mu_y+\mu_z}$.
Another way to think about this is label the time spent in each state as $X_1+Y_1+Z_1+X_2+Y_2+Z_2$......
The proportion of time spent in state $A$ is the time spent in $A$ divided by the entire time, or $\displaystyle \frac{X_1+X_2...}{X_1+Y_1+Z_1+X_2+Y_2+Z_2...}=\frac{n\mu_x}{n(\mu_x+\mu_y+\mu_z)}$