I just started learning regression, so my background is not very strong. I have to solve a certain problem, which doesn't seem too hard, however I have some trouble in understanding it.
Suppose we define
x<-seq(1,50,1), y<-3*x+2+6*rnorm(length(x))
Hence my response $Y$ is exactly of this type: $Y=X\beta+\epsilon$, where $\epsilon$ is the error and normally distributed (here with mean 0 and variance 1). Now we should do a simple linear regression $n$ times and saving the $n$ slopes of the regression.
reg <-lm(y~x)
The true value of the slope should be $3$, as assumed. I calculated the $n$ slopes and saved them in a vector $s$. The next task is to draw a histogram of the $n$ estimated slopes and add the normal density of the theoretically true distribution of the slopes. Here my problem starts: it is clear that the slopes are normally distributed. Hence I have to calculate the mean and variance. Looking at the model: $y_i=\beta_1+x_{i,2}\beta_2 +\epsilon_i$, where $\epsilon_i\sim\mathcal{N}(0,6^2)$. The hint is to use solve for the inverse of a matrix. I do not see how this should help? Is it meant, that I should calculate this:
$$X^{-1}(Y-\epsilon)=\beta$$
How do I get from there the mean and variance? And what is the difference to the mean and empirical standard deviation of the estimated slopes? Thanks in advance for your help.
