Physics307L F08:Schedule/Week 13 agenda/Weighted/Derivation

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

Probability is

${\displaystyle Prob(x_{A})\propto {\frac {1}{\sigma _{A}}}e^{-{\frac {(x_{A}-X)^{2}}{2\sigma _{A}^{2}}}}}$

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

Simplify with chi-squared short-hand

${\displaystyle Prob(x_{A},x_{B})\propto {\frac {1}{\sigma _{A}\sigma _{B}}}e^{-\chi ^{2}/2}}$
${\displaystyle \chi ^{2}=\left({\frac {x_{A}-X}{\sigma _{A}}}\right)^{2}+\left({\frac {x_{B}-X}{\sigma _{B}}}\right)^{2}}$

Step 4: Solve for X

obtain result on previous page.