Expectation values
Consider
 fi 1  fi    fi 1  fi  
2


 x
3
 x2  x1  x2  P  x1  P  x2  P  x3  dx1dxx dx3
2

The cross terms are odd and hence integrate to zero leaving
 fi 1  fi    fi 1  fi  
2





x32 P  x3  dx3  4  x22 P  x2  dx2 


 x P  x  dx
2
1
1
1

Or
 fi 1  fi    fi 1  fi  
2

 6  x 2 P  x  dx

Then using the results from Integral of a Gaussian.doc, this is
 fi 1  fi    fi 1  fi  
6
2
2 
2 2
N
The N is the number of data points in the region considered. For the assumed form
 2  a  b  f   cf 2 , N is on the order of the number of independent determinations of
the above or ~N/3. Thus the error/sigma2 is ~6/N or on the order of 6/100 = 0.06 for
10,000 points. For a more reasonable 96=16*6 points, the error in 2 is about ¼ ot 2.
This makes it difficult for the error to follow small angles.
Figure 1 Artificial data from ..\ArtDat\for\GENDAT.WPJ using ..\ArtDat\for\dfermi.dir. The error
term is f*(0.01) as described in Estimating the error per point.doc. There are 5000 points in test.out
K:\PUBLIC~1\Fittery\ErrorEstimation>errest
THE CODE ERREST IS DESIGNED TO FIT THE ERROR AS
ERR=SQRT(A+B*ABS(F)+C*F**2)
OPTIONS ARE
1 FIT A ONLY, B AND C ARE ZERO
2 FIT A AND B ONLY C = 0
3 FIT A, B AND C
4 FIT B ONLY
5 FIT C ONLY
ENTER THE DESIRED OPTION
5
ENTER THE NAME OF THE DATA FILE
test.out
A =
0.0000000000000000
B =
0.0000000000000000
C =
0.0116123901542815 – Err/sigma = .16, while 6/5000 = 0.085
c is a bit high.
ERR=
0.0000000000000000
OUTPUT WILL BE IN test.c
K:\PUBLIC~1\Fittery\ErrorEstimation>errest
THE CODE ERREST IS DESIGNED TO FIT THE ERROR AS
ERR=SQRT(A+B*ABS(F)+C*F**2)
OPTIONS ARE
1 FIT A ONLY, B AND C ARE ZERO
2 FIT A AND B ONLY C = 0
3 FIT A, B AND C
4 FIT B ONLY
5 FIT C ONLY
ENTER THE DESIRED OPTION
3
ENTER THE NAME OF THE DATA FILE
test.out
A =
-0.0575477145140940
B =
0.1507880821819592
C =
0.0052752652935748
ERR=
2.7755575615628910D-017
OUTPUT WILL BE IN test.abc
Figure 2 The dots are the error estimates for each point. The two solid plots are a fit using c alone,
and one using a,b, anc c in the form 2 = a + b f + c f2. The 3 constants do not follow the data as well
as the fit using c alone.