# Physics307L:Schedule/Week 13 agenda/Weighted/Derivation

Jump to: navigation, search

## Step 1: Probability of each measured mean, given parent distributions with same true value

### Probability is

$Prob(x_A) \propto \frac{1}{\sigma_A}e^{- \frac{(x_A-X)^2}{2 \sigma_A^2}}$

## Step 2: What is joint probability of getting both means?

### Simplify with chi-squared short-hand

$Prob(x_A, x_B) \propto \frac{1}{\sigma_A \sigma_B}e^{-\chi^2 / 2}$
$\chi^2 = \left( \frac{x_A-X}{\sigma_A}\right)^2 + \left( \frac{x_B-X}{\sigma_B}\right)^2$

## Step 4: Solve for X

obtain result on previous page.