c) Find the Yule-Walker equation for the $$ \mathrm{AR}(2) $$ process $$ X_{t}=\frac{1}{3} X_{t-1}+\frac{2}{9} X_{t-2}+e_{t}, e_{t} \sim W N\left(0, \sigma_{e}^{2}\right) $$

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To derive the **Yule-Walker equations** for the given **AR(2)** process:

$$
X_{t} = \frac{1}{3} X_{t-1} + \frac{2}{9} X_{t-2} + e_{t}, \quad e_{t} \sim \text{WN}\left(0, \sigma_{e}^{2}\right),
$$

we start by recalling the general form of the Yule-Walker equations for an **AR(p)** process. For an **AR(2)** process, the Yule-Walker equations relate the autocovariances $$\gamma(k)$$ to the parameters of the model.

### General Yule-Walker Equations for AR(2)

For an AR(2) process defined as:

$$
X_t = \phi_1 X_{t-1} + \phi_2 X_{t-2} + e_t,
$$

the Yule-Walker equations are:

1. **For lag $$k = 1$$:**
   $$
   \gamma(1) = \phi_1 \gamma(0) + \phi_2 \gamma(1)
   $$

2. **For lag $$k = 2$$:**
   $$
   \gamma(2) = \phi_1 \gamma(1) + \phi_2 \gamma(0)
   $$

3. **For lag $$k = 0$$ (Variance Equation):**
   $$
   \gamma(0) = \phi_1 \gamma(1) + \phi_2 \gamma(2) + \sigma_e^2
   $$

### Applying to the Given AR(2) Process

Given the specific coefficients:
$$
\phi_1 = \frac{1}{3}, \quad \phi_2 = \frac{2}{9},
$$
the Yule-Walker equations become:

1. **For lag $$k = 1$$:**
   $$
   \gamma(1) = \frac{1}{3} \gamma(0) + \frac{2}{9} \gamma(1)
   $$

2. **For lag $$k = 2$$:**
   $$
   \gamma(2) = \frac{1}{3} \gamma(1) + \frac{2}{9} \gamma(0)
   $$

3. **For lag $$k = 0$$ (Variance Equation):**
   $$
   \gamma(0) = \frac{1}{3} \gamma(1) + \frac{2}{9} \gamma(2) + \sigma_e^2
   $$

### Summary of the Yule-Walker Equations

Putting it all together, the Yule-Walker equations for the given AR(2) process are:

$$
\begin{cases}
\gamma(1) = \dfrac{1}{3} \gamma(0) + \dfrac{2}{9} \gamma(1) \
\gamma(2) = \dfrac{1}{3} \gamma(1) + \dfrac{2}{9} \gamma(0) \
\gamma(0) = \dfrac{1}{3} \gamma(1) + \dfrac{2}{9} \gamma(2) + \sigma_e^2
\end{cases}
$$

These equations can then be solved simultaneously to find the autocovariances $$\gamma(0)$$, $$\gamma(1)$$, and $$\gamma(2)$$ in terms of the noise variance $$\sigma_e^2$$.
The Yule-Walker equations for the given AR(2) process are: $$ \begin{cases} \gamma(1) = \frac{1}{3} \gamma(0) + \frac{2}{9} \gamma(1) \ \gamma(2) = \frac{1}{3} \gamma(1) + \frac{2}{9} \gamma(0) \ \gamma(0) = \frac{1}{3} \gamma(1) + \frac{2}{9} \gamma(2) + \sigma_e^2 \end{cases} $$ These equations relate the autocovariances of the process to its parameters.

c) Find the Yule-Walker equation for the process

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