Proofab2.tex

\documentclass{article}
\usepackage[utf8]{inputenc}

\title{Obtaining a and b}
\author{Atalia Navarro Boullosa }
\date{June 2018}

\begin{document}
Show explicitly that the linear least square method is derived from the minimization of the chisquare when the model is a straight line.
\ \\
Beginning with 
\begin{equation}
 \chi^{2}=\sum \frac{(y_{i}-y(x_{i},\theta))^{2}}{\sigma_{y_{i}}^{2}}
\end{equation}
where $\sigma_{y_{i}}^{2}= \sum(y_{i}-\bar{y})^{2}$. \\
If we have $y(x_{i},\theta)=a*x_{i}+b$ and derivate respect to $a$ for minimize 
\begin{equation}
 \frac{\partial \chi^{2}}{\partial a}=\frac{\sum(y_{i}-ax_{i}-b)(-x_{i})}{\sigma_{y_{i}}^{2}}=0
\end{equation}
simplifying $b$ we get
\begin{equation}
b= \frac{\sum x_{i}y_{i}-a\sum x_{i}^{2}}{\sum x_{i}} .
\label{eq:b}
\end{equation}
Derivating respect to $b$ for minimize
\begin{equation}
\frac{\partial \chi^{2}}{\partial a}=\frac{\sum(y_{i}-ax_{i}-b)(-1)}{\sigma_{y_{i}}^{2}}=0
\end{equation}
where $\sum y_{i}-a\sum x_{i} -nb=0$, substituting \ref{eq:b} we get 
\begin{equation}
\sum y_{i}-a\sum x_{i}-n\frac{\sum x_{i}y_{i}-a\sum x_{i}^{2}}{\sum x_{i}}=0 .
\end{equation}
By the other hand we know that 
\begin{equation}
\bar{y}=\sum \frac{y_{i}}{n} ;
\bar{x}=\sum \frac{x_{i}}{n}
\end{equation}
So
\begin{equation}
\frac{a}{\bar{x}}(\sum x_{i}^{2}-\bar{x}\sum x_{i})=\frac{\sum x_{i}y_{i}}{\bar{x}}-\sum y_{i}
\end{equation}
and finally
\begin{equation}
a=\frac{\sum x_{i}y_{i}-n\bar{x}\bar{y}}{\sum x_{i}^{2}-n\bar{x}^{2}}.
\end{equation}
Substituting the last equation for a in \ref{eq:b}
\begin{equation}
b=\frac{\sum x_{i}y_{i}}{\sum x_{i}}-\frac{\sum x_{i}y_{i}-n\bar{x}\bar{y}}{\sum x_{i}^{2}-n\bar{x}^{2}}\frac{\sum x_{i}^{2}}{\sum x_{i}}
\end{equation}
or 
\begin{equation}
b=\frac{1}{n\bar{x}(\sum x_{i}^{2}-n\bar{x}^{2})}(\sum x_{i}y_{i}(\sum x_{i}^{2}-n\bar{x}^{2})-(\sum x_{i}y_{i}-n\bar{x}\bar{y})\sum x_{i}^{2})
\end{equation}
and simplifying 
\begin{equation}
b=\frac{\bar{y}\sum x_{i}^{2}-\bar{x}\sum x_{i}y_{i}}{\sum x_{i}^{2}-n\bar{x}^{2}}
\end{equation}
which give us the result we expected. 
\end{document}