# Time Series (Random Walk?)

#### Asano Kyouya

##### New Member
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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