I have the following linear regression model
$$y = \beta_0 + \beta_1 \cdot 40$$
where $\beta_0 = 11.1317$, $\beta_1 = 1.01$, and $40$ is simply the value of the predictor variable (I guess). As you can see, if I plug the betas in the model, for example the expectation would be $51.5317$ (but I think it's not correct from the values of the expectation that I have. See below).
and I would like to calculate the expected value and the variance, but I am not sure how to do it, even having the rules for computing the expectation and variance. From my slides $$E[y] = 51.7590$$and $$Var[y] = 18$$ but I am getting different results. How would you then calculate the expectation and variance of $y$ step by step?
In general you need more information. To estimate the expection value and the variance of $\hat y$ you can use the following formulas:
$$E(\hat y)=\beta_0+\beta_1 \overline x=\overline y$$
$$var(\hat y)=\frac{1}{n-2}\sum_{i=1}^n(y_i-\hat y_i)^2=\frac{1}{n-2}\sum_{i=1}^n(y_i-\beta_0-\beta_1 x_i)^2$$