I am working on some parameter estimation problems and the following solution was shown:
What I don't understand is how the coefficients were found on the second line. Can some one explain how to find the variance of a point estimator in the least technical way possible? Thank you.
The second line comes from the linearity of variance of independent random variables: $$\mathsf{Var}(aX + b Y) = a^2\mathsf{Var}(X)+b^2\mathsf{Var}(Y)$$
Thus as presumably your $X_1, X_2$ are independent random variables: $$\mathsf{Var}(\tfrac 1 4 X_1 + \tfrac 3 4 X_2) = \tfrac 1{16}\mathsf{Var}(X_1)+\tfrac 9{16}\mathsf{Var}(X_2)$$
From the third line it would also appear to be that the variables $X_1, X_2$ both have the same variance: $\sigma^2$, so: $$\mathsf{Var}(\tfrac 1 4 X_1 + \tfrac 3 4 X_2) = \tfrac 1{16}\sigma^2+\tfrac 9{16}\sigma^2 \\ = \tfrac 5 8 \sigma^2$$
That is all.