How to compute basic statistics of the time series $Y_j = A \sin(\theta \times j) + Z_j$

Question

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:

1. Expected value

$$ \mathbb{E}[Y_j] = \mathbb{E}[A \sin(\theta \times j) + Z_j] = \mathbb{E}[A \sin(\theta \times j)] + \mathbb{E}[Z_j] = \mathbb{E}[A \sin(\theta \times j)] = \dots?\dots $$

2. Variance

$$ \mathbb{V}(Y_j) = \mathbb{V}(A \sin(\theta \times j) + Z_j) = \mathbb{V}(A \sin(\theta \times j)) + \mathbb{V}(Z_j) = \mathbb{V}(A \sin(\theta \times j)) + \sigma^2 = \dots ? \dots $$

3. 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?

Answer 1

Basic notions: Let $X$ be a r.v. with $\mathbb{E} X = \mu$ and $\operatorname{var}(X) = \sigma ^ 2$. Take scalars $a$ and $b$, hence $\mathbb{E} (aX + b) = a\mu + b$ and $\operatorname{var}(aX + b) = a ^ 2 \sigma ^2 $ and $\operatorname{cov}(aX + b, Y) = a \operatorname{cov}(X, Y)$, therefore

1. $ \mathbb{E}[Y_i] = \sin \theta i \times\mathbb{E} A + 0 = 0.$

2. $ \operatorname{var}[Y_i] = \sin ^ 2 \theta i \times 1 + \sigma ^ 2 = \sigma ^ 2 + \sin ^ 2\theta i.$

3. \begin{align} cov(Y_i, Y_j) &= cov(A \sin \theta i + z_{j}, A \sin \theta j + z_j)\\ & = cov(A\sin \theta i , A\sin \theta j) + cov(A\sin \theta i , z_i) + cov(A\sin \theta j , z_i) + cov(z_i , z_j)\\ &=\sin\theta i \sin \theta j\times1 + 0 + 0 + \sigma ^ 2\\ &=\sin\theta i \sin \theta j + \sigma ^ 2 . \end{align}