The discussion centers on solving the equation \(\sum_{i=1}^{n}\frac{1}{\sigma_{i}^{2}}2x(y_{i}-\alpha x -\beta x^{2})=0\) for the variables \(\alpha\) and \(\beta\). Participants clarify that the summation does not include \(x\), only \(\sigma_{i}\) and \(y_{i}\). A unique solution for \(\alpha\) and \(\beta\) cannot be obtained without an additional equation. The conversation concludes with the realization that the summation also involves \(x\), resolving the initial confusion.
PREREQUISITESMathematicians, data scientists, and anyone involved in solving complex equations or optimization problems will benefit from this discussion.
No. If the terms in the sum can be positive or negative, their sum can be zero without any of them being zero. As a simple example, 2 + (-1) + (-1) = 0, but no single term equals zero.Dixanadu said:Hey guys,
Was just wondering something. Suppose I have an equation of the form
\sum_{i=0}^{n}\frac{1}{x_{i}}(a-by_{i})=0,
how would I solve this? do I just set the summand = 0?
What variables are you solving for? What symbols represent known values ?Dixanadu said:how would I solve this?
Dixanadu said:The real equation I'm dealing with is
\sum_{i=1}^{n}\frac{1}{\sigma_{i}^{2}}2x(y_{i}-\alpha x -\beta x^{2})=0
Dixanadu said:The closest I can get is
(\alpha x +\beta x^{2})\sum_{i=1}^{n}\frac{1}{\sigma_{i}^{2}}=\sum_{i=1}^{n}\frac{y_{i}}{\sigma_{i}^{2}}
No idea what to do from here