Transcript p - Saed Dababneh

Counting Statistics and Error Prediction
Poisson Distribution (p << 1)
• Low cross section.
Appendix C
• Weak resonance.
• Short measurement (compared to t1/2).
Success ≡ Birthday today.
p = 1/365.
n = 1000.
HW 25
x  pn
x x




pn e
x e
P( x) 

x!
n
 P( x)  1
x 0
x!
We need to
know only the
product.
n
x   xP( x)  pn
x 0
n
 2    x  x 2 P( x)  pn  x
x 0
Radiation Detection and Measurement, JU, First Semester, 2010-2011
(Saed Dababneh).
1
Counting Statistics and Error Prediction
Gaussian (Normal) Distribution (p << 1,
1
P( x) 
e
2x
x > 20)

x  x 2

2x
n
 P( x)  1
Success ≡ Birthday today.
p = 1/365.
n = 10000.
HW 26
x 0
n
x   xP( x)  pn
x 0
n
 2    x  x 2 P( x)  pn  x
x 0
• Can be expressed as a function of .
• Can be expressed in a continuous form.
Radiation Detection and Measurement, JU, First Semester, 2010-2011
(Saed Dababneh).
2
Counting Statistics and Error Prediction
P( x) 
1
e
2x

x  x 2

2x
2
2 2x
G ( ) 
e
x
Radiation Detection and Measurement, JU, First Semester, 2010-2011
(Saed Dababneh).
3
Counting Statistics and Error Prediction
Calculate the percentage of the
samples that will deviate from the
mean by less than:
• one .
HW 27
• two .
• etc …
Radiation Detection and Measurement, JU, First Semester, 2010-2011
(Saed Dababneh).
4
Counting Statistics and Error Prediction
P( x) 
y  y0 
Baseline
offset
1
e
2x
2x

A
w

x  x 2


e
Total area under the
curve above the baseline
2 ( x  x0 ) 2
w2
2
HW 28
2, approximately 0.849 the
width 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?
Radiation Detection and Measurement, JU, First Semester, 2010-2011
(Saed Dababneh).
5
Counting Statistics and Error Prediction
Applications 1- Match experiment to model
•
•
•
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).
1
s 
N
2
x  xe (all what we need)

xe   xF ( x)
x 0
N
 (x  x )
i 1
i
2
e
and not
1 N
s 
( xi  xe ) 2

N  1 i 1
2
because we set x  xe
Radiation Detection and Measurement, JU, First Semester, 2010-2011
(Saed Dababneh).
6
Counting Statistics and Error Prediction
Back to our example
• We can’t use Gaussian
model for this data set.
Why?
• Qualitative comparison.
• Is 2 close to s2?
1
s 
N
2
N
 (x  x)
i 1
i
2
 7.36
Only to guide
the eye!
HW 29
 2  x  8 .8
• Close!? Less fluctuation
than predicted!
• But quantitatively?
• Chi-squared test.
Radiation Detection and Measurement, JU, First Semester, 2010-2011
(Saed Dababneh).
7
Counting Statistics and Error Prediction
Chi-squared
By definition:
Thus:
or
1
 
xe
2
N
 x  x 
i 1
2
i
e
2
2




N

1
s
N

1
s
2 

xe
2
N  1

2
 19  7.36



15
.
891


8.8


s2
2
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).
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Counting Statistics and Error Prediction
smaller <= Fluctuation => larger
Perfect
fit
15.891 (interpolation) or
http://www.stat.tamu.edu/~west/applets/chisqdemo.html
either gives  = 0.6646
Conclusion:
no abnormal fluctuation.
Radiation Detection and Measurement, JU, First Semester, 2010-2011
(Saed Dababneh).
9
Counting Statistics and Error Prediction
Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed
Dababneh).
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Counting Statistics and Error Prediction
Single measurement
S 2 = 2
≈ x
x  x  x  
 x x
68% probability that this
interval includes the true
average value.
What if we want 99%..?
Fractional standard deviation
x
1

x
x
Need 1%?
Count 10000.
Radiation Detection and Measurement, JU, First Semester, 2010-2011
(Saed Dababneh).
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Counting Statistics and Error Prediction
A series of
“single”
measurements.
Radiation Detection and Measurement, JU, First Semester, 2010-2011
(Saed Dababneh).
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Counting Statistics and Error Prediction
Radiation Detection and Measurement, JU, First Semester, 2010-2011
(Saed Dababneh).
13
Counting Statistics and Error Prediction
• What about derived quantities? (Error propagation).
Net counts = Gross counts – Background
Derived
Measured
Gross counts = 1000
Background counts = 400
Net counts = 600  37
Compare to addition
instead of subtraction.
(Count, stop, count).
(not 600  24)
Count Rate = ?  ?
Radiation Detection and Measurement, JU, First Semester, 2010-2011
(Saed Dababneh).
14
Counting Statistics and Error Prediction
Mean value of multiple independent counts.
• Assume we record N repeated counts from a single source for equal counting
times:
  x1  x2  ...  xN
 2   x2   x2  ...   x2
1
2
N
• For Poisson or Gaussian distributions:
 X  xi
i
So that
  x1  x2  ...  xN  
2

  
Radiation Detection and Measurement, JU, First Semester, 2010-2011
(Saed Dababneh).
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Counting Statistics and Error Prediction
Standard error of the mean

x
N


Nx
x 


N
N
N
x
x 
N
• To improve statistical precision of a given measurement by a factor of two
requires four times the initial counting time.

Radiation Detection and Measurement, JU, First Semester, 2010-2011
(Saed Dababneh).
16
Counting Statistics and Error Prediction
Optimizing counting time.
How to divide the limited available beam time?
S = (net) source count rate.
B = background count rate.
TS+B = time to count source + background.
TB = time to count background.
To minimize s :
TSB
S B

TB optimum
B
HW 30
Radiation Detection and Measurement, JU, First Semester, 2010-2011
(Saed Dababneh).
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Counting Statistics and Error Prediction
• Background measurement?
• Without the “source”.
• Should include all sources
except the “source”.
• Accelerator applications:
background with beam.
• Minimum detectable amount.
• False positives and false negatives.
• Rest of Chapter 3.
Radiation Detection and Measurement, JU, First Semester, 2010-2011
(Saed Dababneh).
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