Given the time series:
$$Y_j = A \sin(\theta \times j) + Z_j$$
where
A random variable with mean $0$ and variance $1.$
Z white noise with mean $0$ and variance $\sigma^2$.
- $\theta \in \left\{0, \pi \right\}$ fixed constant
- with $A$ and $Z$ uncorrelated
I would like to compute its expected value, variance and auto-covariance function, and eventually say if it is a stationary process. I am stuck with the algebra because I lack some basics in probability and statistics.
Here is what I can do:
- Expected value
$$ \mathbb{E}[Y_j] = \mathbb{E}[A \sin(\theta \times j) + Z_j] = \mathbb{E}[A \sin(\theta \times j)] + \mathbb{E}[Z_j] = \mathbb{E}[A \sin(\theta \times j)] = \dots?\dots $$
- Variance
$$ \mathbb{V}(Y_j) = \mathbb{V}(A \sin(\theta \times j) + Z_j) = \mathbb{V}(A \sin(\theta \times j)) + \mathbb{V}(Z_j) = \mathbb{V}(A \sin(\theta \times j)) + \sigma^2 = \dots ? \dots $$
- Auto-covariance
$$ \mathbb{C}(Y_j, Y_{j+l}) = \mathbb{C}(A \sin(\theta \times j) + Z_j, A \sin(\theta \times (j+l)) + Z_{j+l}) = \\ = \mathbb{C}(A \sin(\theta \times j) + Z_j, A \sin(\theta \times j + \theta \times l)) + Z_{j+l}) = \dots ? \dots $$
Could you explain the steps to do to solve 1, 2 and 3?