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Change on Chapter 6 Question 7 #84

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a5f115c
Edited answer to chapter 6 question 7
andylikescodes Aug 1, 2017
4837ec3
edited chapter 6 question 7
andylikescodes Aug 1, 2017
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68 changes: 44 additions & 24 deletions ch6/7.md
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Original file line number Diff line number Diff line change
Expand Up @@ -2,19 +2,39 @@ Chapter 6: Exercise 7
========================================================

### a
From $\epsilon\sim N(0, \sigma^2)$ we know that $Y \sim N(\beta_0+\sum_{j=1}^p\beta_jx_{ij}, \sigma^2)$. The distribution of $Y$ is given by:

$$
f(y\mid X, \beta) =
\frac{
1
}{
\sigma \sqrt{2\pi}
}
\exp
\left(-
\frac{
y - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
}{
2\sigma^2
}\right)
$$



The likelihood for the data is:

$$
\begin{aligned}
L(\theta \mid \beta)
&= p(\beta \mid \theta)
L(Y \mid X, \beta)
&= f(Y \mid X, \beta)
\\
&= p(\beta_1 \mid \theta)
&= f(y_1 \mid X, \beta)
\times \cdots
\times p(\beta_n \mid \theta)
\times f(y_n \mid X,\beta)
\\
&= \prod_{i = 1}^{n}
p(\beta_i \mid \theta)
f(y_i \mid X, \beta)
\\
&= \prod_{i = 1}^{n}
\frac{
Expand All @@ -25,7 +45,7 @@ $$
\exp
\left(-
\frac{
Y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j X_{ij})
y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
}{
2\sigma^2
}
Expand All @@ -48,7 +68,7 @@ $$
}
\sum_{i = 1}^{n}
\left[
Y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j X_{ij})
y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
\right]^2
\right)
\end{aligned}
Expand Down Expand Up @@ -87,7 +107,7 @@ $$
}
\sum_{i = 1}^{n}
\left[
Y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j X_{ij})
y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
\right]^2
\right)
\left(
Expand Down Expand Up @@ -123,7 +143,7 @@ $$
}
\sum_{i = 1}^{n}
\left[
Y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j X_{ij})
y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
\right]^2
-
\frac{
Expand Down Expand Up @@ -177,7 +197,7 @@ $$
}
\sum_{i = 1}^{n}
\left[
Y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j X_{ij})
y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
\right]^2
-
\frac{
Expand Down Expand Up @@ -215,7 +235,7 @@ $$
}
\sum_{i = 1}^{n}
\left[
Y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j X_{ij})
y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
\right]^2
+
\frac{
Expand Down Expand Up @@ -260,7 +280,7 @@ $$
}
\sum_{i = 1}^{n}
\left[
Y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j X_{ij})
y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
\right]^2
+
\frac{
Expand Down Expand Up @@ -289,7 +309,7 @@ $$
}
\sum_{i = 1}^{n}
\left[
Y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j X_{ij})
y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
\right]^2
+
\frac{
Expand All @@ -308,7 +328,7 @@ $$
}
\sum_{i = 1}^{n}
\left[
Y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j X_{ij})
y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
\right]^2
+
\frac{
Expand All @@ -329,7 +349,7 @@ $$
\left(
\sum_{i = 1}^{n}
\left[
Y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j X_{ij})
y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
\right]^2
+
\frac{
Expand All @@ -351,7 +371,7 @@ $$
\,
\sum_{i = 1}^{n}
\left[
Y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j X_{ij})
y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
\right]^2
+
\lambda
Expand Down Expand Up @@ -444,7 +464,7 @@ $$
}
\sum_{i = 1}^{n}
\left[
Y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j X_{ij})
y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
\right]^2
\right)
\left(
Expand Down Expand Up @@ -491,7 +511,7 @@ $$
}
\sum_{i = 1}^{n}
\left[
Y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j X_{ij})
y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
\right]^2
- \frac{
1
Expand Down Expand Up @@ -546,7 +566,7 @@ $$
}
\sum_{i = 1}^{n}
\left[
Y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j X_{ij})
y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
\right]^2
- \frac{
1
Expand Down Expand Up @@ -585,7 +605,7 @@ $$
}
\sum_{i = 1}^{n}
\left[
Y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j X_{ij})
y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
\right]^2
+
\frac{
Expand Down Expand Up @@ -633,7 +653,7 @@ $$
}
\sum_{i = 1}^{n}
\left[
Y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j X_{ij})
y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
\right]^2
+
\frac{
Expand Down Expand Up @@ -663,7 +683,7 @@ $$
}
\sum_{i = 1}^{n}
\left[
Y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j X_{ij})
y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
\right]^2
+
\frac{
Expand All @@ -687,7 +707,7 @@ $$
\left(
\sum_{i = 1}^{n}
\left[
Y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j X_{ij})
y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
\right]^2
+
\frac{
Expand Down Expand Up @@ -717,7 +737,7 @@ $$
\left(
\sum_{i = 1}^{n}
\left[
Y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j X_{ij})
y_i - (\beta_0 + \sum_{j = 1}^{p} \beta_j x_{ij})
\right]^2
+
\lambda
Expand Down