Let $x[n]=0\mathrm{.}9x[n-1]+u[n],$ where $u[n]$ is white, zero$-$mean r$.$p$.$ with variance $1.$

$($a$)$ Is $x[n]$ a wide$-$sense stationary r$.$p$.?$ Explain.

$($b$)$ Determine the auto$-$correlation function of $x[n].$

$($c$)$ Determine the first order LMMSE predictor for $x[n](hat(x)[n]=ax[n-1]).$

$($d$)$ Given a realization of $u[n]=\{ubrace(1$ubrace${)}_n=0,0,-1,2,0\}.$ Obtain the corresponding re$-$

alization of $x[n].$ Assume $x[-1]=0.$

$($e$)$ Estimate $x[5]$ using the predictor obtained in part $($c$)$ and from the predictor ob$-$

tained from the data in $($d$).$ Compare the results of the two predictors.