# 14章章末問題
## 14.1
### 問題
Analysis of radon measurements:　(a) Fit a linear regression to the logarithms of the radon measurements in Table 7.3, with indicator variables for the three counties and for whether a measurement was recorded on the first floor. Summarize your posterior inferences in nontechnical terms.
　(b) Suppose another house is sampled at random from Blue Earth County. Sketch the posterior predictive distribution for its radon measurement and give a 95% predictive interval. Express the interval on the original (unlogged) scale. (Hint: you must consider the separate possibilities of basement or first-floor measurement.)

## 14.4.
### 問題
Ordinary linear regression: derive the formula for s2 in (14.7) for the posterior distribu- tion of the regression parameters.

## 14.7
### 問題
Posterior predictive distribution for ordinary linear regression: show that $p(\tilde{y}|\sigma, y)$ is a　normal density. (Hint: first show that $p(\tilde{y}, \beta|\sigma, y)$ is the exponential of a quadratic form in $(\tilde{y}, \beta)$ and is thus is a normal density.)
### 解答
まず、すべての同時分布を書き出す：
\begin{align}
p(\beta, \log \sigma, y, \tilde{y})=N(y|\beta, \sigma^{2} I)N(\tilde{y}|\beta, \sigma^{2} I)
\end{align}
$y$と$\sigma$を定数とすると、上はそれらの条件付き分布として捉えられる。
したがって上は$(\beta, \tilde{y})$の二次関数である。
$(\beta, \tilde{y})$が同時的に正規分布を持つなら、$\tilde{y}$も同時分布である。