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[Phys-L] Re: Average earlier or average later?

On Sep 9, 2005, at 11:43 PM, Bernard Cleyet wrote:

Ludwik!
pse post the source code.

PROGRAM TESTING
!***********************
randomize
let n=1000 ! how many samples from random distr
let meanx=2.2
let stdev=0.5
let sum=0
print "stdev=";stdev
for i=1 to n
call GET_X(mean,stdev,x) ! "measuring" (get x)
let y=x^4 ! calculating from x
let sum=sum+y
next i
let YY=sum/n ! mean from y
print " YY=";YY
end

SUB GET_X(meanx,stdev,x) ! "measuring one x" from
!********************** ! an imposed Gaussian distr
let sum=0
for i=1 to 12
let sum=sum+rnd
next i
let z=sum-6
let x=z*stdev+meanx
end sub


On Sep 9, 2005, at 11:42 AM, Pamela L. Gay wrote:


Okay I'm game for a simulation. Played with excel for 30 seconds and
for
N = 1000
<X_N> = 2.200
stdev(X_N) = 0.374

I found
Y= Sum over N (X_N)^4 / N = 27.499
Y = <X_N>^4 = 23.423

On Sep 9, 2005, at 11:23 AM, ludwik kowalski wrote:


cut


I do not know how wide was the distribution of T in Pamela's
simulation. It turns out, as intuitively expected, that the
discrepancy
between the true Y and the predicted Y grows rapidly when the
distribution of T becomes wider. This is shown below. I am talking
about averaging at the level of Y; averaging at the level of T
produces
a nearly perfect prediction at any standard deviation. Like Pamela, I
used 1000 simulated measurements for each sigma and the mean T=2.2. My
distributions of T were Gaussian.

stdev predicted Y comment
0.0001 23.426 nearly perfect because . . .
0.1 23.71 not a large systematic error . . .
0.25 25.32
0.50 30.25 even larger systematic error due . . .
0.75 42.39 very big systematic error . . .
1.00 60.40 worse
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