weakly stationary time series - what type of model is this?

Question

Say I have the following model: $y_t = 0.5y_{t−1} +x_t +v_{1t}$, and $x_t = 0.5x_{t−1} +v_{2t}$, where both $v_{1t}$ and $v_{2t}$ follow IID normal distribution ∼ (0,1).

How would I go about showing whether the following statement is true or false?

'If a sample of 1000 observations were simulated from this model for $y_t$, then at least five values will be expected to be larger than 6.'

I have been struggling with this question for many days, is it an old exam question, and I have never seen a question similar to this before... does this type of model have a specific name? I am struggling to find anything online to help me understand.

Answer 1

The first part of the question:

I think you should have calculated the unconditional mean and variances. These are time-constant, because the processes are stationary: $$E(y_t ) = E(y_{t-j} ),$$ and $$var(y_t ) = var(y_{t-j} ),$$ and the same for $x_t$.

Theses are my results: $$E(y_t ) =E(x_t )=0, \quad var(x_t ) = \frac{4}{3}, \quad var(y_t ) = \frac{28}{9} $$

The unconditional standard error of $y_t$ is $1.7$ and is normally distributed. This means that $0.02$ % of observations will be larger than 6. In other words, in a sample of 10000, I expect 2 observations to be larger than 6.