probability theory.pdf (21).pdf

c19g, 5/13/1996 09:22
CHAPTER 19
PHYSICAL MEASUREMENTS
W eha v e seen, in Chapter 7, ho w the great mathematician Leonhard Euler w as unable to solv e the
problem of estimating 8 orbital parameters from 75 discrepan t observ ations of the past p ositions
of Jupiter and Saturn. Thinking in terms of deductiv e logic, he could not ev en conceiv e of the
principles b y whic h suc h a problem could b e solv ed. But 38 y ears later Laplace, thinking in terms
of probabilit y theory as logic, w as in p ossession of exactly the righ t principles to resolv e the Great
Inequalit y of Jupiter and Saturn. In this Chapter w e dev elop the solution as it w ould b e done
to da yb y considering a simpler problem, estimating t w o parameters from three observ ations. But
our general solution, in matrix notation, will include Laplace's automatically .
Reduction of Equations of Condition
Supp ose w e wish to determine the c harge e and mass m of the electron. The Millik an oil-drop
exp erimen t measures e directly . The de
ection of an electron b eam in a kno wn ic
eld measures the ratio e=m . The de
ection of an electron to w ard a metal plate due to attraction
2
of image c harges measures e =m .
F rom the results of an yt w o of these exp erimen ts w e can calculate v alues of e and m . But all the
measuremen ts are sub ject to error, and the v alues of e , m