Consider a time series that is well described by the model $$ X_t = 2 + 0.6 X_{t-1} + 0.2X_{t-2} + \varepsilon_t, $$ where $\varepsilon_t\sim \mathcal N(0,\sigma_\varepsilon^2)$ is a white noice process. Then $E[X_t] = 10$.
My question is: why is the expected value of the time series $10$? What calculations do I need to compute?
First you should convince yourself that to solve this, you need to assume that the time series process given by the representation you have is stationary. You can check whether such assumption makes sense.
Now, if the time series is stationary, then the expectation does not change over time, i.e. $E[X_t] = E[X_{t-k}]$ for all $k$.
Taking the expectation on both sides of your equation then gives
$$ E[X_t] = 2+ 0.6\, \overbrace{E[X_t]}^{ = E[X_{t-1}]}\, + 0.2\,\overbrace{E[X_t]}^{ = E[X_{t-2}]}\, + \overbrace{\,0\,}^{ = E[\varepsilon_t]} $$ which is an equation you can probably easy solve.