The error in forecasting for period $T+1$ at the period $T$ is defined as

$e_{T+1}=x_{T+1}-$hat$(x{)}_T(1),$

where hat$(x{)}_T(1)$ is the one$-$step forecast made at time $T.$

The simple exponential smoothing algorithm is defined by

$S_T=x_T+(1-)S_{T-1}.$

Assume simple exponential smoothing is applied to obtain forecasts for the linear trend process

$x_t=b_1+b_{2}t+lo{n}_t,\{lo{n}_t\}NID(0,{}_{lon}^2).$

Find the variance of the forecast error ${}_e^2$:$=$Var$(e_{T+1}),$ in terms of ${}_{lon}^2$ and $.$