# Physics307L:Schedule/Week 8 agenda/Linear fit theory

Following John R. Taylor, "An Introduction to Error Analysis," 2nd edition, Chapter 8:

### We have a relation as follows, and want to fit $\ A$ and $\ B$ to the data

$\ y=A+Bx$

### For a given $\ A$ and $\ B$, the probability for each $\ y_i$ is:

$Prob(y_i) \propto \frac{1}{\sigma_y}e^{-(y_i-A-Bx_i)^2/2\sigma_y^2}$

### And we can call the probability of getting all of the data points as:

$Prob = Prob(y_1) \cdot Prob(y_2) \cdot ... \cdot Prob(y_N)$

### Each term has the same σy, so can be simplified as:

$Prob \propto \frac{1}{\sigma_y^N}e^{-\chi^2/2}$
$chi-squared, \chi^2 = \sum_{i=1}^N \frac{\left (y_i - A - Bx_i \right )^2}{\sigma_y^2}$

### To maximize the probability, minimize the chi-squared sum ... take derivatives, solve system of equations, obtain:

(1) $AN + B \sum x_i = \sum y_i$

(2) $A \sum x_i + B \sum x_i^2 = \sum x_i y_i$

$A=\frac{\sum x_i^2 \sum y_i^2 - \sum x_i \sum x_i y_i}{\Delta}$
$B=\frac{N\sum x_i y_i - \sum x_i \sum y_i}{\Delta}$
$\Delta=N \sum x_i^2 - \left ( \sum x_i \right )^2$