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

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## See Taylor 2nd edition page 174 or Bevington 2nd edition page 58

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

### (I.e., assume both data sets have the same "true" value, but with differing standard deviations)

### Assume Guassian distributions

### Probability is

- [math]\displaystyle{ Prob(x_A) \propto \frac{1}{\sigma_A}e^{- \frac{(x_A-X)^2}{2 \sigma_A^2}} }[/math]

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

### Simplify with chi-squared short-hand

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

## Step 3: Principle of maximum likelihood: minimize chi-squared with respect to X

## Step 4: Solve for X

obtain result on previous page.