I have question on the picture
I will answer first two i.e. (a) and (b) questions because eNotes rules forbid answering multiple questions.
Because of linearity of expectation `E[a X+bY]=aE[x]+bE[Y]`
By definition `Var(X)=E[X^2]-(E[X])^2` hence we have
Here we've used expected value from part (a) and now we will use linearity of expectation.
` ` Now because `beta^2E[x_i^2]-beta^2(E[x_i])^2=beta^2Var(x_i)`
` `and `E[epsilon_i^2]=Var(epsilon_i)-(E[epsilon_i])^2=sigma^2-0^2=sigma^2` we have
` ` `beta^2Var(x_i)+2betaE[x_iepsilon_i]+sigma^2` which would be your solution.
If we assume that `E[x_iepsilon_i]=0` which will be the case if the estimator `beta` is unbiased and consistent, then you would have
`beta^2Var(x_i)+sigma^2` <-- Your solution if `x_i` and `epsilon_i` are uncorrelated.