Let where are the coefficients to be determined.
Then apply the orthogonality principle
Solve for the three coefficients using the 3 equations.
After computing the coefficients, compute
for the MMSE.
You didn't learn about it in class? Do you have a formula involving matrix inversion?
In its simplest form, the Orthogonality Principle states that "the error vector of the optimum estimator is orthogonal to any other possible estimator."
I assumed that you have had some linear algebra and understood the concept of inner product. If two vectors are orthogonal, then its inner product is zero, ie .
Here, the inner product of two random variables and is defined to be .
Now let the optimum estimator be where are the coefficients to be determined. You can think of as three linearly independent vectors(like the cartesian basis , although I should add that the u's are not orthogonal like the cartesian basis)
The error of the optimum estimator is thus .
Now apply the orthogonality principle, and we have
where is any possible estimator.
We have three unknowns to solve for so we need three linearly independent equations. They can be obtained by substituting with (each of the u's is a possible estimator by itself). So the three equations are
Substitute in the expression for , expand out, plug in the values, and solve for the 3 unknown coefficients and you're done.