Physics307L F07:Schedule/Week 14 agenda/Weighted/Derivation

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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

(See p. 174 of Taylor)

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

Step 4: Solve for X

obtain result on previous page.

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