1. Prove that if ten points (xi, yi) all lie on a parabola (y=a2(x^2)+a1x+a0), then the least-squares
solution to find the best-fit parabola is in fact this parabola. (hint: show what the normal
equations might look like in general, and show that (a0, a1, a2) is guaranteed to be a solution.)
Solution
y =a0+a1x+a2(x^2)
The normal equations are
sum(y) =n*a0+a1*sum(x)+a2*sum(x^2) (1)
sum(y*x) = a0*sum(x)+a1*sum(x^2)+a2*sum(x^3) (2)
sum(y*x^2)= a0*sum(x^2)+a1*sum(x^3)+a2*sum(x^4) (3)
These three equations are the normal equations. Here the only unknown quantities are a0, a1 and
a2. All other quantities, can be find out from the data given ( xi, yi) , i=1,2, ... , n. (Note that the
mathe matical simbol used for 'sum' is 'sigma'). Since, there are 3 equations in three unknown
quantities (a0, a1 and a2), these simultenous equations can be solved using any standard mathe
mathematical technique to obtain (a0, a1, a2). So the solution is guaranteed