PRML 第1章 1.6(基本)

---

$$
\begin{array}{ll}
(1.41) & \mathrm{cov}[x,y] = \mathbb{E}_{x,y} \bigl[ \{x-\mathbb{E}[x]\} \{y-\mathbb{E}[y]\} \bigr] = \mathbb{E}_{x,y}[xy] - \mathbb{E}[x] \mathbb{E}[y]
\end{array}
$$

---

解答

変数$${x}$$，$${y}$$が独立であるとき，$${p(x,y) = p(x)p(y)}$$が成り立つ．したがって，

・離散変数の場合

$$
\begin{array}{lll}
\mathbb{E}_{x,y}[xy] &=& {\displaystyle \sum_{x}\sum_{y}} xy~p(x,y)\\[1.5em]
&=& {\displaystyle \sum_{x}\sum_{y}} xy~p(x) p(y)\\[1.5em]
&=& {\displaystyle \sum_x} xp(x) {\displaystyle \sum_y} yp(y)\\[1.5em]
&=& \mathbb{E}[x] \mathbb{E}[y]
\end{array}
$$

・連続変数の場合

$$
\begin{array}{lll}
\mathbb{E}_{x,y}[xy] &=& {\displaystyle \iint} xy~p(x,y) dxdy\\[1em]
&=& {\displaystyle \iint} xy~p(x) p(y) dxdy\\[1em]
&=& {\displaystyle \int} xp(x)dx {\displaystyle \int} yp(y)dy\\[1em]
&=& \mathbb{E}[x] \mathbb{E}[y]
\end{array}
$$

となり，

$$
\mathrm{cov}[x,y] = \mathbb{E}_{x,y}[xy] - \mathbb{E}[x] \mathbb{E}[y] = 0
$$

が成り立つ．