回復 13# ilikemath 的帖子
9.
兩組二維數據\((u_i,v_i)\)與\((x_i,y_i)\)，變量\(U\)與\(X\)、\(V\)與\(Y\)分別有線性關係如下：\(u_i=-x_i+7\)、\(\displaystyle v_i=\frac{1}{2}y_i+2\)，已知\(V\)對\(U\)的迴歸直線通過點\((8,3)\)且\(\mu_x=4\)、\(\mu_y=6\)則\(Y\)對\(X\)的迴歸直線方程式為\(y=\) 。
[解答]
\(v\) 對 \(u\)的迴歸直線通過(8, 3) \(\rightarrow\) 設\(v=a(u-8)+3\)
由題目說的線性關係得\(\displaystyle\frac{1}{2}y+2=a(-x+7-8)+3\)
\(\rightarrow\) \(y=-2a(x+1)+2\)
又此直線必過\((\mu_x, \mu_y)=(4, 6)\) 〈題目出得不好，沒講清楚這兩個分別是\(x,y\)的平均〉
解得\(\displaystyle a=\frac{-2}{5}\) \(\rightarrow\) \(\displaystyle y=\frac{4}{5}(x+1)+2\)
