Simple linear regression

\text{Var}(Y_i - \bar{Y}) = \text{Var}(Y_i) + \text{Var}(\bar{Y})- 2\text{Cov}(Y_i,\bar{Y}) = \sigma^2 + \dfrac{\sigma^2}{n} - 2\text{Cov}(Y_i, \bar{Y})

Since, in simple linear regression

\text{Var}(Y_i) = \sigma^2

and

\text{Var}(\bar{Y}) = \dfrac{\sigma^2}{n}

I'm not sure how to find

\text{Cov}(Y_i, \bar{Y})

.
I started by doing

\text{Cov}(Y_i, \dfrac{1}{n}\sum_{i=1}^n Y_i) = \dfrac{1}{n}\text{Cov}(Y_i, \sum_{i=1}^n Y_i)

, as 1/n is some constant. But i also in simple linear regression

\text{Cov}(Y_i, Y_i) = \text{Var}(Y_i) = \sigma^2

.

Don't know what to do from here on as i have that summation involved.
Any help would be great!

A-levels and AS-levels, explained

What is an LLM and how could it benefit your career?

Powerful questions business students forget to ask their career advisers

Why I chose a law degree and what studying law was like