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