Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh). 1 Counting Statistics and Error Prediction isson Distribution (p << 1) x pn x P x x pn x xP x x P x e x x e pn x P n x n x n x x x pn x 0 2 2 0 0 ) ( ) ( 1 ) ( ! ! ) ( Success ≡ Birthday today. p = 1/365. n = 1000. Low cross section. Weak resonance. Short measurement (compared to t 1/2 ). Appendix C We need to know only the product. HW 25 HW 25 Asymmetric
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Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh). 1 Counting Statistics and Error Prediction Poisson Distribution ( p.
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Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).
1
Counting Statistics and Error PredictionPoisson Distribution (p << 1)
xpnxPxx
pnxxPx
xP
x
ex
x
epnxP
n
x
n
x
n
x
xxpnx
0
22
0
0
)(
)(
1)(
!!)(
Success ≡ Birthday today.p = 1/365.n = 1000.
• Low cross section.• Weak resonance.• Short measurement (compared to t1/2).
Appendix C
We need to know only the
product.
HW 25HW 25
Asymmetric
Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).
• Can be expressed as a function of .• Can be expressed in a continuous form.
HW 26HW 26
Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).
3
Counting Statistics and Error Prediction
x
xx
ex
xP 2
2
2
1)(
xex
G 2
2
2)(
Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).
4
Counting Statistics and Error PredictionCalculate the percentage of the samples that will deviate from the mean by less than:• one .• two .• etc …
HW 27HW 27
Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).
5
2
20 )(2
2
0w
xx
ew
Ayy
x
xx
ex
xP 2
2
2
1)(
Counting Statistics and Error Prediction
Baselineoffset
Total area under the curve above the baseline
22, approximately 0.849 the, approximately 0.849 thewidth of the peak at half heightwidth of the peak at half height
This model describes a bell-shaped curve like the normal (Gaussian) probability distribution function. The center x0 represents the "mean", while ½ w is the standard deviation.
What is FWHM? Resolution? Peak centroid? What is FWHM? Resolution? Peak centroid?
HW 28HW 28
Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).
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Counting Statistics and Error PredictionApplicationsApplications 1- Match experiment to model
N
iei xx
Ns
1
22 )(1
1
0
)(x
e xxFx
need) what we(all exx
N
iei xx
Ns
1
22 )(1
and not
because we set exx
• Assume a specific distribution (Poisson, Gaussian).
• Set distribution mean to be equal to experimental mean.
• Compare variance to determine if distribution is valid for actual data set (Chi-squared test).
Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).
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• We can’t use Gaussian model for this data set. Why?• Qualitative comparison.• Is 2 close to s2?
• Close!? Less fluctuation than predicted! • But quantitatively?• Chi-squared test.
Counting Statistics and Error Prediction
36.7)(1
1
22
N
ii xx
Ns
Only to guide the eye!
8.82 x
Back to our example
HW 29HW 29
By definition:
Thus:
or
Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).
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Counting Statistics and Error Prediction
N
iei
e
xxx 1
22 1
Chi-squaredChi-squared
891.15
8.8
36.719112
222
sN
x
sN
e
2
22
1 s
N
The degree to which 2 differs from (N-1) is a measure of the departure of the data from predictions of the distribution.
Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).