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```Intermediate Lab
PHYS 3870
Lecture 3
Distribution Functions
References: Taylor Ch. 5 (and Chs. 10 and 11 for Reference)
Taylor Ch. 6 and 7
Also refer
to0 “Glossary
Important
Terms in Error Analysis”
Introduction
Section
Lecture 1 ofSlide
1
“Probability Cheat Sheet”
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 1
Intermediate Lab
PHYS 3870
Distribution Functions
Introduction
Section 0
Lecture 1
Slide 2
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 2
Practical Methods to Calculate Mean and St. Deviation
We need to develop a good way to tally, display, and think about
a collection of repeated measurements of the same quantity.
Here is where we are headed:
• Develop the notion of a probability distribution function, a distribution to
describe the probable outcomes of a measurement
• Define what a distribution function is, and its properties
• Look at the properties of the most common distribution function, the Gaussian
distribution for purely random events
• Introduce other probability distribution functions
We will develop the mathematical basis for:
•
•
•
•
•
•
•
•
•
•
Mean
Standard deviation
Standard deviation of the mean (SDOM)
Moments and expectation values
Error propagation formulas
Introduction Section 0 Lecture 1 Slide 3
Schwartz inequality (i.e., the uncertainty principle) (next lecture)
Numerical values for confidence limits (t-test)
INTRODUCTION TO Modern Physics PHYX 2710
Principle
of maximal
likelihood
Fall 2004
Central limit theorem
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 3
Two Practical Exercises in Probabilities
Flip a
Penny
penny5050
times
times
and
andrecord
record
the
the
results
results
Roll a pair of dice 50 times and record the results
Introduction
Section 0
Lecture 1
Slide 4
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Grab a partner and a
set of instructions
and complete the
exercise.
Lecture 3 Slide 4
Two Practical Exercises in Probabilities
Flip a penny 50 times and record the results
What isIntroduction
the asymmetry
of the
Section 0 Lecture
1 results?
Slide 5
INTRODUCTION TO Modern Physics PHYX 2710
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Intermediate 3870
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 5
Two Practical Exercises in Probabilities
Flip a penny 50 times and record the results
??% asymmetry
4% asymmetry
What isIntroduction
the asymmetry
of the
Section 0 Lecture
1 results?
Slide 6
INTRODUCTION TO Modern Physics PHYX 2710
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Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 6
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value?
The standard deviation?
Introduction
Section 0
Lecture 1
Slide 7
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Intermediate 3870
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 7
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value?
The standard deviation?
What is the asymmetry
Introduction Section 0
(kurtosis)?
Lecture 1
Slide 8
What is the probability of
rolling a 4?
INTRODUCTION TO Modern Physics PHYX 2710
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 8
Discrete Distribution Functions
Written in terms of “occurrence” F
A data set to play with
In terms of fractional expectations
The mean value
Fractional expectations
Normalization condition
Introduction
Section 0
Lecture 1
Slide 9
Mean value

(This is just a weighted sum.)
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 9
Limit of Discrete Distribution Functions
Binned data sets
“Normalizing” data sets
fk ≡ fractional occurrence
∆k ≡ bin width
Mean value: 𝑋 =
𝐹𝑘 𝑥𝑘 =
𝑘
𝑘
Introduction
Normalization:
(𝑓𝑘 Δ𝑘 )𝑥𝑘
Section𝑘 (𝑓
0 𝑘 ΔLecture
1
𝑘 )𝑥𝑘
1=
Slide 10
𝑘 Δ𝑘
𝐹4
= 𝑓4
𝑁
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Expected value:
Fall 2004
𝑃𝑟𝑜𝑏
4 =
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 10
Limits of Distribution Functions
Consider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Introduction
Section 0
Lecture 1
Slide 11
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 11
Continuous Distribution Functions
Meaning of Distribution Interval
Thus
= fraction of measurements
that fall within a<x<b
Normalization of Distribution

and by extension
Central (mean) Value
Introduction

Section 0
Width of Distribution
Lecture 1
Slide 12

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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 12
Moments of Distribution Functions
The first moment for a probability distribution function is
𝑥 ≡ 𝑥 = 𝑓𝑖𝑟𝑠𝑡 𝑚𝑜𝑚𝑒𝑛𝑡 =
+∞
𝑥
−∞
𝑓(𝑥) 𝑑𝑥
For a general distribution function,
𝑥 ≡ 𝑥 = 𝑓𝑖𝑟𝑠𝑡 𝑚𝑜𝑚𝑒𝑛𝑡 =
+∞
𝑥 𝑔(𝑥)𝑑𝑥
−∞
+∞
𝑔(𝑥)𝑑𝑥
−∞
Generalizing, the nth moment is
𝑥𝑛 ≡ 𝑥
𝑛
= 𝑛𝑡ℎ 𝑚𝑜𝑚𝑒𝑛𝑡 =
+∞ 𝑛
𝑥 𝑔(𝑥)𝑑𝑥
−∞
+∞
𝑔(𝑥)𝑑𝑥
−∞
=
+∞
−∞
𝑥 𝑛 𝑓(𝑥) 𝑑𝑥
0 (for a centered distribution)
th
Introduction
O moment
≡ NSection 0
1st moment ≡ 𝑥
Lecture
1 Slide 13≡
2nd moment
rd
𝑥−𝑥
2
→ 𝑥2
3 moment ≡ kurtosis
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 13
Moments of Distribution Functions
Generalizing, the nth moment is
𝑥𝑛 ≡ 𝑥
𝑛
= 𝑛𝑡ℎ 𝑚𝑜𝑚𝑒𝑛𝑡 =
+∞ 𝑛
−∞ 𝑥 𝑔(𝑥)𝑑𝑥
+∞
𝑔(𝑥)𝑑𝑥
−∞
=
+∞
−∞
2nd moment ≡ 𝑥 − 𝑥 2
3rd moment ≡ kurtosis
Oth moment ≡ N
1st moment ≡ 𝑥
𝑥 𝑛 𝑓(𝑥) 𝑑𝑥
0
→ 𝑥2
The nth moment about the mean is
𝜇𝑛 ≡
=
𝑥−𝑥
+∞
−∞
𝑛
= 𝑛𝑡ℎ 𝑚𝑜𝑚𝑒𝑛𝑡 𝑎𝑏𝑜𝑢𝑡 𝑡ℎ𝑒 𝑚𝑒𝑎𝑛
𝑥−𝑥 𝑛 𝑔(𝑥)𝑑𝑥
+∞
−∞
𝑔(𝑥)𝑑𝑥
=
+∞
−∞
𝑥−𝑥
𝑛
𝑓(𝑥) 𝑑𝑥
The standard deviation (or second moment about the mean) is
Introduction
𝜎𝑥2 ≡ 𝜇2 ≡
+∞
−∞
Section 0
𝑥−𝑥
Lecture 1
2
𝑥−𝑥 2 𝑔(𝑥)𝑑𝑥
= 2𝑛𝑑 𝑚𝑜𝑚𝑒𝑛𝑡 𝑎𝑏𝑜𝑢𝑡 𝑡ℎ𝑒 𝑚𝑒𝑎𝑛
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=
+∞Fall 2004
𝑔(𝑥)𝑑𝑥
−∞
Intermediate 3870
Fall 2013
Slide 14
=
+∞
−∞
𝑥−𝑥
2
𝑓(𝑥) 𝑑𝑥
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 14
Example of Continuous Distribution Functions and Expectation Values
Harmonic Oscillator: Example from Mechanics

Expected Values
The expectation value of a function Ɍ(x) is
Ɍ x
≡
=
+∞
Ɍ
−∞
+∞
−∞
+∞
−∞
x 𝑔 𝑥 𝑑𝑥
𝑔 𝑥 𝑑𝑥
Ɍ(x) 𝑓(𝑥) 𝑑𝑥
Introduction
Section 0
Lecture 1
Slide 15
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 15
Example of Continuous Distribution Functions and Expectation Values
Boltzmann Distribution: Example from Kinetic Theory
Expected Values

The expectation value of a function
Ɍ(x) is
Ɍ x
≡
=
+∞
Ɍ
−∞
+∞
−∞
+∞
−∞
x 𝑔 𝑥 𝑑𝑥
The Boltzmann distribution function for velocities of particles as
a function of temperature, T is:
𝑔 𝑥 𝑑𝑥
𝑀
𝑃 𝑣; 𝑇 = 4𝜋
2𝜋 𝑘𝐵 𝑇
Ɍ(x) 𝑓(𝑥) 𝑑𝑥
3 2
𝑣 2 𝑒𝑥𝑝
1
𝑀𝑣 2
2
1
𝑘 𝑇
2 𝐵
Then
v =
v
Introduction
Section 0
2
Lecture 1
=
+∞
−∞
𝑣 𝑃(𝑣) 𝑑𝑣 =
+∞
−∞
v
2
8 𝑘𝐵 𝑇
𝑃(𝑣) 𝑑𝑣 =
1 2
𝜋𝑀
3 𝑘𝐵 𝑇
Slide 16
1 2
𝑀
3
implies KE = 12𝑀 v 2 = 𝑘𝐵 𝑇
2
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vpeak=
2 𝑘𝐵 𝑇
1 2
𝑀
1 2
= 2 3
DISTRIBUTION FUNCTIONS
v2
Lecture 3 Slide 16
Example of Continuous Distribution Functions and Expectation Values
Fermi-Dirac Distribution: Example from Kinetic Theory
For a system of identical fermions, the average number of fermions in a single-particle state , is given
by the Fermi–Dirac (F–D) distribution,
where kB is Boltzmann's constant, T is the absolute temperature,
state , and μ is the total chemical potential.
is the energy of the single-particle
Since the F–D distribution was derived using the Pauli exclusion principle, which allows at most one
electron to occupy each possible state, a result is that
When a quasi-continuum of energies has an associated density of states
(i.e. the number of
states per unit energy range per unit volume) the average number of fermions per unit energy range
per unit volume is,
where
is called the Fermi function
Introduction
Section 0
Lecture 1
Slide 17
so that,
INTRODUCTION TO Modern Physics PHYX 2710
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 17
Example of Continuous Distribution Functions and Expectation Values
Finite Square Well: Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is O x
2
For a finite square well of width L, Ψn x =
Ψn ∗ x |Ψn x
≡
∗
𝑥 = Ψn x |x|Ψn x
+∞
Ψn ∗ x 𝑂
−∞
+∞
∗
Ψ
x
n
−∞
≡
L sin
𝑥 Ψ x 𝑑𝑥
Ψ x 𝑑𝑥
nπx
L
𝑥 Ψn x 𝑑𝑥
Ψn x 𝑑𝑥
+∞
∗
Ψ
x 𝑥 Ψn x
n
−∞
+∞
∗
Ψ
x Ψn x
n
−∞
≡
+∞ ∗
Ψ x 𝑂
−∞
+∞
Ψ∗ x
−∞
=1
𝑑𝑥
𝑑𝑥
= 𝐿/2
+∞
ℏ ∂
∗
Ψ
x
Ψn x 𝑑𝑥
n
ℏ
∂
−∞
i
∂x
∗
𝑝 = Ψn x |
|Ψn x ≡
=0
+∞
∗
i
∂x
Introduction Section 0 Lecture 1 Slide 18 −∞ Ψn x Ψn x 𝑑𝑥
𝐸𝑛 = Ψn ∗ x |iℏ
∂
|Ψ x
∂t n
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≡
+∞
∗
Ψ
x
n
−∞
+∞
∗
Ψ
n
−∞
∂
Ψ x 𝑑𝑥 𝑛2 𝜋 2 ℏ2
∂t n
=
2 𝑚 𝐿2
x Ψn x 𝑑𝑥
iℏ
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 18
Gaussian Integrals
Factorial Approximations
𝑛! ≈ 2𝜋𝑛
1 2
𝑛𝑛 𝑒𝑥𝑝 −𝑛 + 121 𝑛 + 𝑂
1
𝑛2
1
1
𝑙𝑜𝑔 𝑛! ≈ 12 𝑙𝑜𝑔 2𝜋 + 𝑛 + 2 log 𝑛 − 𝑛 + 12 𝑛 + 𝑂
1
𝑛2
𝑙𝑜𝑔 𝑛! ≈ 𝑛 log 𝑛 − 𝑛 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
𝐼𝑚 = 2
∞ 𝑚
𝑥
0
exp⁡
(−𝑥 2 ) 𝑑𝑥
𝐼𝑚 = 2
∞ 𝑛
𝑦
0
exp⁡
(−𝑦) 𝑑𝑦 ≡ Γ(𝑛 + 1)
1
𝐼0 = Γ 𝑛 = 2 =
𝜋
Introduction
; m>-1
; 𝑥 2 ≡ 𝑦, 2 𝑑𝑥 = 𝑦 1
; m=0, 𝑛 = −12
Section 0
Lecture 1
2
𝑑𝑦,
𝑛 ≡ 12 (𝑚 − 1)
Slide 19
1
𝐼2 𝑘 = Γ 𝑘 + 2 = 𝑘 − 1 2 𝑘 − 3 2 . . . 3 2 1 2
𝜋
; even m m=2 k>0, 𝑛 = 𝑘 − 12
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𝐼2 𝑘+1 = Γ 𝑘 + 1 = 𝑘!
Fall 2004
Intermediate 3870
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; odd m m=2 k+1>0, 𝑛 = 𝑘 ≥ 0
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 19
Intermediate Lab
PHYS 3870
The Gaussian Distribution
Function
Introduction
Section 0
Lecture 1
Slide 20
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 20
Gaussian Distribution Function
Independent
Variable
Center of Distribution
(mean)
Distribution
Function
Normalization
Constant
Width of Distribution
(standard deviation)
Gaussian Distribution Function
Introduction
Section 0
Lecture 1
Slide 21
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 21
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N ∞ and dx0
Introduction
Section 0
Lecture 1
Slide 22
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 22
Gaussian Distribution Moments
Add proofs of normalization, mean and SD notes
Introduction
Section 0
Lecture 1
Slide 23
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 23
Defining the Gaussian
distribution function
Introduction
Section 0
Lecture 1
Slide 24
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 24
define other common
distribution functions.
Consider the limiting
distribution function
as N ȸ and ∆k0
Introduction
Section 0
Lecture 1
Slide 25
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 25
When is mean x not Xbest?
distribution is not
Example: Cauchy
Distribution
Introduction
Section 0
Lecture 1
Slide 26
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 26
When is mean x not Xbest?
Introduction
Section 0
Lecture 1
Slide 27
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 27
When is mean x not Xbest?
distribution is has
more than one peak.
Introduction
Section 0
Lecture 1
Slide 28
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 28
Intermediate Lab
PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
Introduction
Section 0
Lecture 1
Slide 29
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 29
Probabilities in Combination
AND
1 Four
P(H,4) = P(H) * P(4)
1 Six
OR
1 Four
P(H,4) = P(H) + P(4)
(true for a “mutually exclusive” single role)
OR
1 Four
P(H,4) = P(H) + P(4) - P(H and 4)
NOT 1 Six
P(NOT 6) = 1 - P(6)
(true for a “non-mutually exclusive” events)
Introduction
Section 0
Lecture 1
Slide 30
Probability of a data set of N like measurements, (x1,x2,…xN)
P (x1,x2,…xN) = P(x)1*P(x2)*…P(xN)
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 30
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for:
• Mean
• Standard deviation
• Standard deviation of the mean (SDOM)
• Moments and expectation values
• Error propagation formulas
measurements)
• Numerical values for confidence limits (t-test)
• Principle of maximal likelihood
• Central limit theorem
Introduction Section 0 Lecture 1 Slide 31
• Weighted
distributions and Chi squared
• Schwartz inequality (i.e., the uncertainty principle) (next lecture)
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 31
Standard Deviation of Gaussian Distribution
See Sec. 10.6: Testing of
Hypotheses
5 ppm or ~5σ “Valid for HEP”
1% or ~3σ “Highly Significant”
5% or ~2σ “Significant”
1σ “Within errors”
Area under curve
(probability that
Introduction Section 0
–σ<x<+σ) is 68%
Lecture 1
Slide 32
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More complete Table in App. A and B
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 32
Error Function of Gaussian Distribution
Error Function: (probability that –tσ<x<+tσ ).
Complementary Error Function:
(probability that –x<-tσ AND x>+tσ ).
Prob(x outside tσ) = 1 - Prob(x within tσ)
Probable Error: (probability that
–0.67σ<x<+0.67σ) is 50%.
Error Function: Tabulated values-see App. A.
Area under curve
(probability that
Section 0
–tσ<x<+tσ) isIntroduction
given
in Table at right.
Lecture 1
Slide 33
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More complete Table in App. A and B
Intermediate 3870
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 33
Useful Points on Gaussian Distribution
Full Width at Half Maximum
Points of Inflection
Occur at ±σ
(See Prob. 5.13)
FWHM
(See Prob. 5.12)
Introduction
Section 0
Lecture 1
Slide 34
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DISTRIBUTION FUNCTIONS
Lecture 3 Slide 34
Error Analysis and Gaussian Distribution
Introduction
XX+A
Section 0
Multiplying by a Constant
Lecture 1
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Slide 35
XBX and σB σ
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 35
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error in Z:
Multiple two probabilities
𝑃𝑟𝑜𝑏 𝑥, 𝑦 = 𝑃𝑟𝑜𝑏 𝑥 ∙ 𝑃𝑟𝑜𝑏 𝑥 ∝ 𝑒𝑥𝑝
∝ 𝑒𝑥𝑝
Introduction
Section 0
Lecture 1
Slide 36
1
−
2
∝ 𝑒𝑥𝑝
1
−
2
(𝑥+𝑦)2
𝜎𝑥 2 +𝜎𝑦 2
(𝑥+𝑦)2
−𝑧 2
𝜎𝑥 2 +𝜎𝑦 2
ICBST (Eq. 5.53)
𝑒𝑥𝑝
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2
X+YZ
and Fallσ2004
x + σ y  σz
Intermediate 3870
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DISTRIBUTION FUNCTIONS
1 𝑥2 𝑦2
−
+
2 𝜎𝑥 2 𝜎𝑦 2
𝑧2
−
2
Integrates to √2π
Lecture 3 Slide 36
General Formula for Error Propagation
General formula for error propagation see [Taylor, Secs. 3.5 and 3.9]
Uncertainty as a function of one variable [Taylor, Sec. 3.5]
1. Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs. x.
c) On the graph, label:
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation:
 q 
q hi  qbest  slope x  qbest   
 x 
 q 
qIntroduction
 x  qbest 0  Lecture

1
low  q best  slope Section
 x 
Slide 37
e) Therefore:
q 
q
 x
x
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Note the absolute value.
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 37
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
q
[Taylor, Eq. 3.23]
q 
 x
x
Can you now derive for specific rules of error propagation:
1. Addition and Subtraction [Taylor, p. 49]
2. Multiplication and Division [Taylor, p. 51]
3. Multiplication by a constant (exact number) [Taylor, p. 54]
4. Exponentiation (powers) [Taylor, p. 56]
Introduction
Section 0
Lecture 1
Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 38
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor, Sec. 3.11]
1. It can easily (no, really) be shown that (see Taylor Sec. 3.11) for a
function of several variables
q( x, y , z ,...) 
q
q
q
 x 
 y 
 z  ...
x
y
z
[Taylor, Eq. 3.47]
2. More correctly, it can be shown that (see Taylor Sec. 3.11) for a
function of several variables
q( x, y , z,...) 
q
q
q
 x 
 y 
 z  ...
x
y
z
[Taylor, Eq. 3.47]
where the equals sign represents an upper bound, as discussed above.
3. For a function of several independent and random variables
Introduction
Section 0
Lecture 1
 q

q( x, y, z,...)  
 x 
 x

2
Slide 39
 q





y
 y



2
 q


 z 
 z

2
 ...
[Taylor, Eq. 3.48]
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Again, the proof is left for Ch. 5.
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 39
Error Propagation: General Case
Consider the arbitrary derived quantity
q(x,y) of two independent random variables x and y.
Expand q(x,y) in a Taylor series about the expected values of x and y
(i.e., at points near X and Y).
Fixed, shifts peak of distribution
𝜕𝑞
𝜕𝑞
𝑞 𝑥, 𝑦 = 𝑞 𝑋, 𝑌 +
𝑥−𝑋 +
(𝑦 − 𝑌)
𝜕𝑥 𝑋
𝜕𝑦 𝑌
Fixed
Distribution centered at X with width σX
Product of Two Variables
Introduction
Section 0
𝛿𝑞 𝑥, 𝑦 = 𝜎𝑞 =
Lecture 1
Slide 40
𝑞 𝑋, 𝑌 +
𝜕𝑞
𝜕𝑥
2
𝑋
𝜎𝑥
+
𝜕𝑞
𝜕𝑦
2
𝜎𝑦
𝑌
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
0
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 40
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as
the square root of the
number of measurements.
Introduction Section 0 Lecture
Central
Limit Theorem
1
Slide 41
That is, the distribution
gets narrower as more
For random, independent measurements (each with a well-define
expectation value and well-defined variance), the arithmetic mean
(average) will be approximately normally distributed.
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 41
Mean of Gaussian Distribution as “Best Estimate”
Principle of Maximum Likelihood
To find the most likely value of the mean (the best estimate of ẋ),
find X that yields the highest probability for the data set.
Consider a data set
{x1, x2, x3 …xN }
Each randomly distributed with
The combined probability for the full data set is the product
∝
1 −(𝑥 −𝑋)2
𝑒 1
𝜎
2𝜎
1
2
× 𝑒 −(𝑥 2 −𝑋)
𝜎
2𝜎
1
2
× … × 𝑒 −(𝑥 𝑁 −𝑋)
𝜎
2𝜎
=
1
𝑒
𝜎𝑁
−(𝑥 𝑖 −𝑋)2 2𝜎
Slide 42
BestIntroduction
EstimateSection
of X0is Lecture
from 1maximum
probability or minimum summation
Solve for
Minimize
Sum INTRODUCTION TO Modern Physics PHYX 2710derivative
Fall 2004
set to 0
Intermediate 3870
Fall 2013
Best
estimate
of X
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 42
Uncertaity of “Best Estimates” of Gaussian Distribution
Principle of Maximum Likelihood
To find the most likely value of the mean (the best estimate of ẋ),
find X that yields the highest probability for the data set.
{x1, x2, x3 …xN }
Consider a data set
The combined probability for the full data set is the product
1 −(𝑥 −𝑋)2
∝ 𝑒 1
𝜎
2𝜎
1 −(𝑥 −𝑋)2
× 𝑒 2
𝜎
2𝜎
1 −(𝑥 −𝑋)2
× …× 𝑒 𝑁
𝜎
2𝜎
1
= 𝑁𝑒
𝜎
−(𝑥 𝑖 −𝑋)2 2𝜎
Best Estimate of X is from maximum probability or minimum summation
Minimize
Sum
Introduction
Section 0
Solve for
derivative
Lecture
1 set
Slide
wrst X
to 043
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Solve for
MinimizeINTRODUCTION TO Modern Physics PHYX 2710
See
derivative
Fall 2004
Sum
wrst σ set to 0 Prob. 5.26
Intermediate 3870
Fall 2013
Best
estimate
of σ
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 43
Intermediate Lab
PHYS 3870
Introduction
Combining Data Sets
Weighted Averages
Section 0
Lecture 1
Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 44
Weighted Averages
Question: How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty?
Introduction
Section 0
Lecture 1
Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 45
Weighted Averages
Consider two measurements of the same quantity, described by a random Gaussian distribution
<x1>  x1
and <x2>  x2
Assume negligible systematic errors.
The probability of measuring two such measurements is
𝑃𝑟𝑜𝑏𝑥 𝑥1 , 𝑥2 = 𝑃𝑟𝑜𝑏𝑥 𝑥1 𝑃𝑟𝑜𝑏𝑥 𝑥2
1 −𝜒 2 /2
𝑒
𝑤ℎ𝑒𝑟𝑒 𝜒 2 ≡
𝜎1 𝜎2
=
𝑥1 − 𝑋
𝜎1
2
+
𝑥2 − 𝑋
𝜎2
2
To find the best value for X, find the maximum Prob or minimum X2
Note: X2 , or Chi squared, is the sum of the squares of the deviations from the mean, divided by
the corresponding uncertainty.
Introduction
Section 0
Lecture 1
Slide 46
Such methods are called “Methods of Least Squares”. They follow directly from the Principle of
Maximum Likelyhood.
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 46
Weighted Averages
The probability of measuring two such measurements is
𝑃𝑟𝑜𝑏𝑥 𝑥1 , 𝑥2 = 𝑃𝑟𝑜𝑏𝑥 𝑥1 𝑃𝑟𝑜𝑏𝑥 𝑥2
=
1 −𝜒 2 /2
𝑒
𝑤ℎ𝑒𝑟𝑒 𝜒 2 ≡
𝜎1 𝜎2
𝑥1 − 𝑋
𝜎1
2
+
𝑥2 − 𝑋
𝜎2
2
To find the best value for X, find the maximum Prob or minimum X2
Best Estimate of χ is from maximum probibility or minimum summation
Solve for derivative wrst χ set to 0
Minimize Sum
Solve for best estimate of χ
𝑥𝑊_𝑎𝑣𝑔 =
Introduction
𝑤1 𝑥1 + 𝑤2 𝑥2
=
𝑤1 + 𝑤2
Section 0
Lecture 1
𝑤𝑖 𝑥𝑖
𝑤ℎ𝑒𝑟𝑒 𝑤𝑖 = 1 𝜎
𝑤𝑖
𝑖
2
Slide 47
Note: If w1=w2, we recover the standard result Xwavg= (1/2) (x1+x2)
Finally, the width of a weighted average distribution is
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
𝜎𝑤𝑖𝑒𝑔 ℎ𝑡𝑒𝑑 𝑎𝑣𝑔 =
1
𝑖 𝑤𝑖
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 47
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or, indeed, data of
related measured and calculated quantities) is Kalman filtering.
The Kalman filter, also known as
is an algorithm that uses a series
of measurements observed over
time, containing noise (random
variations)
and
other
inaccuracies,
and
produces
estimates of unknown variables
that tend to be more precise than
those
based
on
a
single
measurement alone.
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
Introduction
0 Lecture
Slide transition
48
estimate. The
estimate Section
is updated
using 1a state
model and measurements. xk|k-1 denotes the
th
estimate of the system's state at time step k before the k measurement yk has been taken into account; Pk|k1 is the corresponding uncertainty. --Wikipedia, 2013.
INTRODUCTION TO Modern Physics PHYX 2710
Ludger Scherliess, of Fall
USU
2004 Physics, is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather .
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 48
Intermediate Lab
PHYS 3870
Introduction
Rejecting Data
Chauvenet’s Criterion
Section 0
Lecture 1
Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 49
Rejecting Data
Chauvenet’s Criterion
Question: When is it “reasonable” to discard a seemingly
“unreasonable” data point from a set of randomly distributed
measurements?
• Never
• Whenever it makes things look better
• Chauvenet’s
criterion
provides
Introduction Section
0 Lecture
1 Slide 50 a (quantitative) compromise
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 50
Rejecting Data
Zallen’s Criterion
Question: When is it “reasonable” to discard a seemingly
“unreasonable” data point from a set of randomly distributed
measurements?
• Never
Introduction
Section 0
Lecture 1
Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 51
Rejecting Data
Disney’s Criterion
Question: When is it “reasonable” to discard a seemingly
“unreasonable” data point from a set of randomly distributed
measurements?
• Whenever it makes things look better
Disney’s First Law
Wishing will make it so.
Introduction
Section 0
Lecture 1
Slide 52
Disney’s Second Law
INTRODUCTION TO Modern Physics PHYX 2710
Dreams are more colorful than reality.
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 52
Rejecting Data
Chauvenet’s Criterion
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50%.
Consider a set of N measurements of a single quantity
{ x1,x2, ……xN }
Calculate <x> and σx and then determine the fractional deviations from the mean of all
the points:
For the suspect point(s), xsuspect, find the probability of such a point occurring in N
measurements
Introduction Section 0 Lecture 1 Slide 53
n = (expected number as deviant as xsuspect)
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004σ )
= N Prob(outside xsuspect·
x
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 53
Error Function of Gaussian Distribution
Error Function: (probability that –tσ<x<+tσ ).
Error Function: Tabulated values-see App. A.
Area under curve
(probability that
Section 0
–tσ<x<+tσ) isIntroduction
given
in Table at right.
Lecture 1
Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Probable Error: (probability
Fall 2004that
–0.67σ<x<+0.67σ) is 50%.
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 54
Chauvenet’s Criterion
Introduction
Section 0
Lecture 1
Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 55
Chauvenet’s Criterion—Example 1
Introduction
Section 0
Lecture 1
Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 56
Chauvenet’s Criterion—Example 1
Introduction
Section 0
Lecture 1
Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 57
Chauvenet’s Criterion—Example 2
Introduction
Section 0
Lecture 1
Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 58
Chauvenet’s
Criterion—
Example 2
Introduction
Section 0
Lecture 1
Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 59
Chauvenet’s
Criterion—
Example 2
Introduction
Section 0
Lecture 1
Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 60
Intermediate Lab
PHYS 3870
Summary of Probability Theory
Introduction
Section 0
Lecture 1
Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 61
Summary of
Probability
Theory-I
Introduction
Section 0
Lecture 1
Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 62
Summary of
Probability
Theory-II
Introduction
Section 0
Lecture 1
Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 63
Summary of Probability Theory-III
Introduction
Section 0
Lecture 1
Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 64
Summary of Probability Theory-IV
Introduction
Section 0
Lecture 1
Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
DISTRIBUTION FUNCTIONS
Lecture 3 Slide 65
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