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

From OpenWetWare

## Contents

- 1 See Taylor 2nd edition page 174 or Bevington 2nd edition page 58
- 2 Step 1: Probability of each measured mean, given parent distributions with same true value
- 3 Step 2: What is joint probability of getting both means?
- 4 Step 3: Principle of maximum likelihood: minimize chi-squared with respect to X
- 5 Step 4: Solve for X
- 6 Step 5: Error propagation to obtain new sigma

## 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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle Prob(x_A, x_B) \propto \frac{1}{\sigma_A \sigma_B}e^{-\chi^2 / 2}}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \chi^2 = \left( \frac{x_A-X}{\sigma_A}\right)^2 + \left( \frac{x_B-X}{\sigma_B}\right)^2}**

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

## Step 4: Solve for X

obtain result on previous page.