1. Easy simple regression model

Let $\displaystyle Y_i=\beta_0+\beta_1X_i+\epsilon_i$

I want to show that $\displaystyle E[b_0]=\beta_0$, where $\displaystyle b_0$ is an estimate of $\displaystyle \beta_0$

$\displaystyle b_0=\overline{Y}-b_1\overline{X}$

$\displaystyle E[b_0]=E[\overline{Y}]-\overline{X}E[b_1]$

$\displaystyle E[b_0]=\beta_0+\overline{X}\beta_1-\overline{X}\beta_1$

$\displaystyle E[b_0]=\beta_0$

Is this valid?

2. Originally Posted by noob mathematician
Let $\displaystyle Y_i=\beta_0+\beta_1X_i+\epsilon_i$

I want to show that $\displaystyle E[b_0]=\beta_0$, where $\displaystyle b_0$ is an estimate of $\displaystyle \beta_0$

$\displaystyle b_0=\overline{Y}-b_1\overline{X}$

$\displaystyle E[b_0]=E[\overline{Y}]-\overline{X}E[b_1]$

$\displaystyle E[b_0]=\beta_0+\overline{X}\beta_1-\overline{X}\beta_1$

$\displaystyle E[b_0]=\beta_0$

Is this valid?
You have not provided enough context. As it is it looks like you have effectivly assumed the answer, or at least imported other assumptions without telling us what they are.

You say $\displaystyle b_0$ is an estimate, but you appear to have a specific extimator in mind.

CB