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

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.

Step 5: Error propagation to obtain new sigma