To solve this problem, let's break down the steps in detail.

Step 1: Understanding the Quadratic Form in the Exponential

We start with the centered normal distribution for $(X_1, X_2)$, with density defined in the form given by equation (29.10), where the exponential has a quadratic form. We're asked to express the quadratic form as: $\frac{1}{\sigma_{11}} x_1^2 + \frac{\sigma_{11}}{D} \left( x_2 - \frac{\sigma_{12}}{\sigma_{11}} x_1 \right)^2$

This expression likely comes from transforming the joint distribution of $(X_1, X_2)$ into a form that separates the contributions of $x_1$ and $x_2$, making it easier to isolate and integrate out $x_2$.

Step 2: Integrate Out $x_2$

We want to integrate $x_2$ out from the joint density function $f(x_1, x_2)$ to obtain the marginal distribution of $X_1$. The problem states: $f(x_1, x_2) = \frac{1}{\sqrt{2 \pi \tau}} \exp\left[ -\frac{1}{2\tau} \left( x_2 - \frac{\sigma_{12}}{\sigma_{11}} x_1 \right)^2 \right]$ where $\tau = \sigma_{22} - \sigma_{12}^2 \sigma_{11}^{-1}$.

The integral to perform here is: $\int_{-\infty}^{\infty} f(x_1, t) \, dt$ This integral would give us the conditional distribution of $X_2$ given $X_1$.

Step 3: Describe the Conditional Distribution of $X_2$ Given $X_1$

The form of the exponential term suggests that $X_2 | X_1$ is normally distributed with mean: $\mathbb{E}[X_2 | X_1] = \frac{\sigma_{12}}{\sigma_{11}} x_1$ and variance: $\text{Var}(X_2 | X_1) = \tau = \sigma_{22} - \frac{\sigma_{12}^2}{\sigma_{11}}$

Thus, the conditional distribution of $X_2$ given $X_1$ is: $X_2 | X_1 \sim \mathcal{N} \left( \frac{\sigma_{12}}{\sigma_{11}} X_1, \sigma_{22} - \frac{\sigma_{12}^2}{\sigma_{11}} \right)$

Would you like further details on any specific step?

Follow-up Questions

1. How is the covariance matrix used to define the joint distribution of two normal variables?
2. How do we derive conditional distributions from joint normal distributions?
3. What is the role of the determinant $D$ in normal distribution problems?
4. Why is integrating out a variable useful in probability density functions?
5. How does the value of $\sigma_{12}$ affect the relationship between $X_1$ and $X_2$?

Tip: When working with conditional distributions in multivariate normals, remember that conditioning often reduces the variance, as seen here with $\tau$.