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\begin{center}
 {\Large Homework 2}
\end{center}


\medskip
\medskip
Let $X_t$ satisfy
$$
\varphi(B)X_t=\theta(B)Z_t,\;\;-\infty<t<\infty
$$
where $\{Z_t\}$ is White Noise $(0,\sigma^2)$.
Find the representation
$$
X_t=\sum_{i=0}^\infty\psi_iZ_{t-i}\;\;\mbox{(casual)}
$$
and
$$
Z_t=\sum_{j=0}^\infty\pi_iX_{t-i}\;\;\mbox{(invertable)}
$$
in the following cases:\\
\begin{enumerate}
\item
$$\varphi(z)=1-z+\frac{1}{4}z^2\;\;\mbox{and}\;\;\theta(z)=1.  $$
\item
$$ \varphi(z)=1+\frac{1}{9}z^2\;\;\mbox{and}\;\;\theta(z)=1+\frac{1}{3}z.  $$
\item
$$ \varphi(z)=(1-\frac{1}{3}z)(1-\frac{1}{2}z)(1-\frac{1}{4}z) \;\;\mbox{and}\;\;\theta(z)=1.  $$
\item
$$ \varphi(z)=(1+\frac{1}{2}z)(1-\frac{1}{5}z) \;\;\mbox{and}\;\;\theta(z)=(1-\frac{1}{2}z)(1-\frac{1}{3}z) $$
\item
$$ \varphi(z)=(1+\frac{1}{2}z)^3 \;\;\mbox{and}\;\;\theta(z)=1.  $$
\end{enumerate}

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