Time Series (Random Walk?)

#1
I just recently started to learn about moving average process of order 1, however, I get confused if there are other things attached to the equation.

1. For example: \(Z_t = 8 + 2t + 5X_t\) where \(X_t\) is a zero-mean stationary series with autocovariance function \(r_k\)

a) Find the mean function and the autocovariance function of \(Z_t\).

I am guessing:

Mean = \(E[Z_t]=E[8+2t+5X_t]=E[8]+2E[t]+E[X_t]=8+2\mu_t\)

Covariance = \(Cov[Z_t,Z_t]=E[Z_tZ_t]=E[8+2t+5X_t][8+2t+5X_t]=r_k^2\)

Can someone please tell me if I did this correct?


2. The questions says \(X_t\) is a zero-mean, unit variance, stationary process with autocorrelation function \(p_k\).

Then how do I find the mean, variance, and auto covariance of \(Z_t = 8 + 2t + 4tX_t\)?


Mean = \(E[Z_t]=E[8+2t+4tX_t]=E[8]+2E[t]+4tE[X_t]=0\) ?

Variance = No idea

Covariance = I guess I just have to use

\(corr(x,x)=cov(x,x)/\sqrt{var(x)var(x)}?\)
 
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