(5) Consider a process satisfying $Y_{t}=\varphi Y_{t- 1}+\varepsilon_{t}$, where $\varphis can be any number, and $\left\ {\varepsilon_{t} ight\}$ is $\mathrm{WN}\left(0, \sigma_{\varepsilon}^{2} ight) $ such that each $\varepsilon_{t}$ is independent of the past $\left\{Y_{t-1}, Y_{t-2}, \ldots, Y_{0} ight\}$. Let $Y_{0}$ be a random variable with mean $\mu_{0}$ and variance $\sigma_{0}^{2}$. SS4861: Show that, for $t \geq 1$, $$ Y_{t}=\varepsilon_{t}+\varphi \varepsilon_{t-1}+\cdots+\varphi^{t-1] \varepsilon_{1}+\varphi^{t} Y_{0} $$ SP. ASO23s