# Showing that $\operatorname{cov}(y_{x_0}, \hat{y_{x_0}}) = 0$

Exercise :

For the Simple Linear Model $$\mathbb E[y_x] = b_0 + b_1x$$, prove that for a newly given $$x_0$$ and $$y_{x_0}$$ a new observation while $$\hat{y_{x_0}}$$ its point estimate, it is : $$\operatorname{cov}(y_{x_0},\hat{y_{x_0}}) = 0$$

Question :

It is :

$$\operatorname{cov}(y_{x_0},\hat{y_{x_0}}) = \mathbb E[(y_{x_0}-\mathbb E[y_{x_0}])(\hat{y_{x_0}}-\mathbb E[\hat{y_{x_0}}])]$$

but how would one continue to prove the expression asked from that point on ?

• Did you search the site? There are several posts on this topic, one of which likely has the answer. – StubbornAtom Nov 1 '18 at 19:50
• @StubbornAtom found none regarding this specific one on either. The fact that covariance operator is not standard in latex makes it hard to find posts. – Rebellos Nov 1 '18 at 20:11