第 14 題：
設數學分數為 \(x_1, x_2, ..., x_n\)，物理分數為 \(y_1, y_2, ..., y_n \)，
兩科加總分的分數為 \(z_1, z_2, ..., z_n\)，兩科的相關係數為 \(r\)，
則 \(\displaystyle \mu_x = 60, \mu_y = 70, \sigma_x = 5, \sigma_y = 6, \sigma_z = 9\)，且 \(\mu_z = \mu_x + \mu_y\)
利用 \(\displaystyle z_i^2 = \left(x_i+y_i\right)^2, i =1, 2, 3, ..., n\) ，
得 \(\displaystyle z_1^2+z^2+ ... +z_n^2 = x_1^2 + x^2+ ... + x_n^2 + 2\left(x_1 y_1+x_2 y_2+...+x_n y_n\right)+y_1^2+y_2^2+...+y_n^2\)
\(\displaystyle \Rightarrow n\left(\mu_z^2 + \sigma_z^2\right) = n\left(\mu_x^2 + \sigma_x^2\right)+2\left(n\sigma_x\sigma_y r+ n\mu_x \mu_y\right) +n\left(\mu_y^2 + \sigma_y^2\right)\)
\(\displaystyle \Rightarrow \sigma_z^2= \sigma_x^2+2\sigma_x\sigma_y r +\sigma_y^2\)
\(\displaystyle \Rightarrow 9^2= 5^2 + 2\cdot 5 \cdot 6 \cdot r +6^2\)
得 \(\displaystyle r=\frac{1}{3}\) 。
故迴歸直線為 \(\displaystyle y-70 = \frac{1}{3}\cdot\frac{6}{5}\left(x-60\right)\)
註1： 相關係數 \(\displaystyle r = \frac{x_1y_1 + x_2 y_2 + ... +x_n y_n - n\mu_x \mu_y}{n\sigma_x\sigma_y}\)
\(\displaystyle \Rightarrow x_1y_1 + x_2 y_2 + ... +x_n y_n = n\sigma_x\sigma_y r +n\mu_x \mu_y\)
註2： \(\displaystyle \mu_z = \mu_x + \mu_y\Rightarrow \mu_z^2 = \mu_x^2+2\mu_x\mu_y+\mu_y^2\)
