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 to Glossary of Important Terms in Error Analysis Introduction Section 0 Lecture 1 Slide 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

Lecture 1 Slide 3 independent and random measurements) Addition of errors in0quadrature (for Schwartz inequality (i.e., the uncertainty principle) (next lecture) Numerical values for confidence limits (t-test) Principle of maximal likelihood Central limit theorem INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 3 Two Practical Exercises in Probabilities Flip a Penny penny 5050 times times and and record 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 Grab a partner and a set of instructions and complete the exercise. Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 4 Two Practical Exercises in Probabilities Flip a penny 50 times and record the results What is the asymmetry of the results? Introduction Section 0 Lecture 1 Slide 5 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS

Lecture 3 Slide 5 Two Practical Exercises in Probabilities Flip a penny 50 times and record the results ??% asymmetry 4% asymmetry What is the asymmetry of the results? Introduction Section 0 Lecture 1 Slide 6 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 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 INTRODUCTION TO Modern Physics PHYX 2710

Fall 2004 Intermediate 3870 Fall 2013 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 (kurtosis)?Introduction Section 0 Lecture 1 Slide 8 What is the probability of rolling a 4? INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 8 Discrete Distribution Functions A data set to play with Written in terms of occurrence F 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.) INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 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: = = ( )

Introduction Normalization: ( ) 0 Lecture 1Section = 1 Slide 10 4 = 4 INTRODUCTION TO Modern Physics PHYX 2710 Expected value: 4 = Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 10 Limits of Distribution Functions

Consider the limiting distribution function as N and k0 Larger data sets Mathcad Games: Introduction Section 0 Lecture 1 Slide 11 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 11 Continuous Distribution Functions Meaning of Distribution Interval Thus = fraction of measurements that fall within a

Introduction Section 0 Width of Distribution Lecture 1 Slide 12 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 12 Moments of Distribution Functions The first moment for a probability distribution function is = + = () For a general distribution function, +

= = () + () Generalizing, the nth moment is + = th O moment N Introduction Section 0 1st moment = () +

() + = () 0 (for a centered distribution) 2 2 2 moment Lecture 1 Slide 13 3rd moment kurtosis nd INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 13 Moments of Distribution Functions Generalizing, the nth moment is +

= = () + () + = () 2 2nd moment 3rd moment kurtosis Oth moment N 1st moment 0 2 The nth moment about the mean is =

+ () = + () + () = The standard deviation (or second moment about the mean) is Introduction 2 Section 0 Lecture 1 2 Slide 14 2 = 2

+ = 2 () + + 2 () = INTRODUCTION TO Modern Physics PHYX 2710 Fall() 2004 Intermediate 3870 Fall 2013 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 (x) is + x x + + = (x) () Introduction Section 0 Lecture 1 Slide 15 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 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 (x) is + x x The Boltzmann distribution function for velocities of particles as a function of temperature, T is: + + = (x) () ; = 4 2 32 1 2 2 2

1 2 Then + v = 2 v Introduction Section 0 Lecture 1 () = + 2 = v 8 () =

12 3 12 3 Slide 16 = 12v2 = implies KE 2 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 vpeak= 2 12 12 = 2 3 Intermediate 3870 Fall 2013 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 FermiDirac (FD) distribution, where kB is Boltzmann's constant, T is the absolute temperature, state , and is the total chemical potential. is the energy of the single-particle Since the FD 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 Fall 2004 Intermediate 3870 Fall 2013 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 + x x + xx nx For a finite square well of width L, n x = 2 L sin L + n x |n x n x n x +

n x n x =1 + n x |x|n x = n x n x + n x n x = /2 + x n x n i x n x | =

| x =0 + i x n n x n x Introduction Section 0 Lecture 1 Slide 18 Physics PHYX 2710 INTRODUCTION TOnModern x |i = Fall 2004 t |n x + n x i t n x + n x n x Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS 222 = 2 2

Lecture 3 Slide 18 Summary of Distribution Functions Available on web site Introduction Section 0 Lecture 1 Slide 19 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 19 Intermediate Lab PHYS 3870 The Gaussian Distribution Function Introduction Section 0 Lecture 1 Slide 20

References: Taylor Ch. 5 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 20 Gaussian Integrals Factorial Approximations 1 ! 2 12 + 121 + 2 1 1 1 ! 12 2 + + 2 log + 12 + 2 ! log (for all terms decreasing faster than linearly with n) Gaussian Integrals = 2 0 exp (2) ; m>-1 ; 2 , 2 = 12 , 12 ( 1) = 2 0 exp () ( + 1)

1 0 = = 2 = Introduction ; m=0, = 12 Section 0 Lecture 1 1 Slide 21 2 = + 2 = 1 2 3 2... 3 2 1 2 2 +1 = + 1 = ! ; even m m=2 k>0, = 12 ; odd m m=2 k+1>0, = 0 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 21 Gaussian Distribution Function Distribution Function Independent Variable

Normalization Constant Center of Distribution (mean) Width of Distribution (standard deviation) Gaussian Distribution Function Introduction Section 0 Lecture 1 Slide 22 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 22 Effects of Increasing N on Gaussian Distribution Functions Consider the limiting distribution function as N and dx0 Introduction Section 0

Lecture 1 Slide 23 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 23 Defining the Gaussian distribution function in Mathcad I suggest you investigate these with the Mathcad sheet on the web site Introduction Section 0 Lecture 1 Slide 24 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS

Lecture 3 Slide 24 Using Mathcad to define other common distribution functions. Consider the limiting distribution function as N and k0 Introduction Section 0 Lecture 1 Slide 25 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 25 Gaussian Distribution Moments Consider the Gaussian distribution function Use the normalization condition to evaluate the normalization constant (see Taylor, p. 132) The mean, , is the first moment of the Gaussian distribution

function (see Taylor, p. 134) Introduction Section 0 Lecture 1 Slide 26 The standard deviation, x, is the standard deviation of the mean of the Gaussian INTRODUCTION TO Modern Physics PHYX 2710 distribution function (see Taylor, p. 143) Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 26 When is mean x not Xbest? Answer: When the distribution is not symmetric about X. Example: Cauchy Distribution Introduction Section 0 Lecture 1 Slide 27

INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 27 When is mean x not Xbest? Introduction Section 0 Lecture 1 Slide 28 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 28 When is mean x not Xbest? Answer: When the distribution is has more than one peak.

Introduction Section 0 Lecture 1 Slide 29 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 29 Intermediate Lab PHYS 3870 The Gaussian Distribution Function and Its Relation to Errors Introduction Section 0 Lecture 1 Slide 30 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870

Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 30 A Review of Probabilities in Combination 1 head AND 1 Four P(H,4) = P(H) * P(4) 1 Six OR 1 Four P(6,4) = P(6) + P(4) (true for a mutually exclusive single role) 1 head 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 31 Probability of a data set of N like measurements, (x 1,x2,xN) P (x1,x2,xN) = P(x)1*P(x2)*P(xN)

INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 31 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 Addition of errors in quadrature (for independent and random measurements) Numerical values for confidence limits (t-test) Principle of maximal likelihood Central limit theorem Introduction

Section 0 Lectureand 1 Slide Weighted distributions Chi32 squared Schwartz inequality (i.e., the uncertainty principle) (next lecture) INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 32 Gaussian Distribution Moments Consider the Gaussian distribution function Use the normalization condition to evaluate the normalization constant (see Taylor, p. 132) The mean, , is the first moment of the Gaussian distribution function (see Taylor, p. 134) Introduction Section 0 Lecture 1 Slide 33 The standard deviation, x, is the standard deviation of the mean of the Gaussian INTRODUCTION TO Modern Physics PHYX 2710

distribution function (see Taylor, p. 143) Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 33 Standard Deviation of Gaussian Distribution See Sec. 10.6: Testing of Hypotheses 5 ppm or ~5 Valid for HEP 1% or ~3 Highly Significant 5% or ~2 Significant 1 Within errors Area under curve (probability that

More complete Table in App. A and B Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 34 Error Function of Gaussian Distribution Error Function: (probability that t+t ). Prob(x outside t) = 1 - Prob(x within t) Probable Error: (probability that 0.67

DISTRIBUTION FUNCTIONS Lecture 3 Slide 35 Useful Points on Gaussian Distribution Full Width at Half Maximum Points of Inflection FWHM (See Prob. 5.12) Introduction Section 0 Occur at (See Prob. 5.13) Lecture 1 Slide 36 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 36 Error Analysis and Gaussian Distribution Adding a Constant

Introduction XX+A Section 0 Multiplying by a Constant Lecture 1 INTRODUCTION TO Modern Physics PHYX 2710 Slide 37 XBX and B Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 37 Error Propagation: Addition Sum of Two Variables Consider the derived quantity Z=X + Y (with X=0 and Y=0) Error in Z: Multiple two probabilities , =

1 2 Introduction Section 0 Lecture 1 Slide 38 (+)2 2+2 1 (+)2 2 2 2+2 ICBST (Eq. 5.53) 2 2 X+YZ and x2 + y2 z2 (addition in quadrature for random, independent variables!) INTRODUCTION TO Modern Physics PHYX 2710 1 2 2 2 2+ 2

Integrates to 2 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 38 General Formula for Error Propagation How do we determine the error of a derived quantity Z(X,Y,) from errors in X,Y,? 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 + x) x) (3) qlow = q(xbest- x) x) d) Making a linear approximation: q q hi qbest slope x qbest x q qlow Introduction qbest slope xSection qbest 0 Lecture 1 x

Slide 39 e) Therefore: q q x xINTRODUCTION TO Modern Physics PHYX 2710 Note the absolute value. Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 39 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 40 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 40 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 2 Slide 41 q q( x, y , z,...) x x q y y 2 q

z z 2 ... [Taylor, Eq. 3.48] INTRODUCTION TO Modern Physics PHYX 2710 Again, the proof is left for Ch. 5. Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 41 Error Propagation: General Case How do we determine the error of a derived quantity Z(X,Y,) from errors in X,Y,? 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 , = , + + ( ) Fixed

Distribution centered at X with width X Product of Two Variables Introduction Section 0 Lecture 1 Slide 42 2 , = = , + + 2 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 0 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 42 SDOM of Gaussian Distribution Standard Deviation of the Mean Each measurement has similar Xi=

and similar partial derivatives Thus The SDOM decreases as the square root of the number of measurements. Introduction Section 0 Lecture 1 Slide 43 That is, the relative width, /, of the distribution gets narrower as more , of the distribution gets narrower as more measurements are made. INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 43 Two Key Theorems from Probability Central Limit Theorem For random, independent measurements (each with a well-define expectation value and well-defined variance), the arithmetic mean (average) will be approximately normally distributed. Principle of Maximum Likelihood Given the N observed measurements, x1, x2,xN, the best estimates for and are those values for which the observed x1, x2,xN, are most likely. Introduction

Section 0 Lecture 1 Slide 44 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 44 Mean of Gaussian Distribution as Best Estimate Principle of Maximum Likelihood To find the most likely value of the mean (the best estimate of ), 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 ( )22 1 ( )22 1 1 2 2 1

2 () 2 = () 2 Introduction Section Slide 45 Best Estimate of X0isLecture from1 maximum probability or minimum summation Solve for Minimize derivative wrst INTRODUCTION TO Modern Physics PHYX 2710 Sum X set to 0 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Best estimate of X Lecture 3 Slide 45 Uncertainty of Best Estimates of Gaussian Distribution Principle of Maximum Likelihood To find the most likely value of the standard deviation (the best estimate of

the width of the x distribution), find x that yields the highest probability for the data set. Consider a data set {x1, x2, x3 xN } The combined probability for the full data set is the product 1 ( )22 1 ( )22 1 ( )22 1 ( )22 1 2 = Best Estimate of X is from maximum probability or minimum summation Minimize Sum Introduction Section 0 Solve for derivative wrst1 X set Lecture Slide to 460 Best estimate

of X Best Estimate of is from maximum probability or minimum summation Minimize Sum INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Solve for See derivative wrst set to 0 Prob. 5.26 Best estimate of Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 46 Intermediate Lab PHYS 3870 Introduction Combining Data Sets Weighted Averages Section 0 Lecture 1 Slide 47

References: Taylor Ch. 7 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 47 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 48 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 48 Weighted Averages

Consider two measurements of the same quantity, described by a random Gaussian distribution x1 Assume negligible systematic errors. and x2 The probability of measuring two such measurements is 1,2 = 1 2 1 2 2 2 1 2/2 2 = + 12 1 2 To find the best value for , find the maximum Prob or minimum 2

Note: 2 , 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 49 Such methods are called Methods of Least Squares. They follow directly from the Principle of Maximum Likelihood. INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 49 Weighted Averages The probability of measuring two such measurements is 1,2 = 1

2 1 2 2 2 1 2/2 2 = + 12 1 2 To find the best value for , find the maximum Prob or minimum 2 Best Estimate of is from maximum probibility or minimum summation Solve for derivative wrst set to 0 Minimize Sum This leads to _ = Introduction Section 0 11+22 1+2 Lecture 1 =

Solve for best estimate of = 1 2 Slide 50 Note: If w1=w2, we recover the standard result X wavg= (1/, of the distribution gets narrower as more 2) (x1+x2) Finally, the width of a weighted average distribution is INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 1 = Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 50 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 linear quadratic estimation (LQE), 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 estimate. The estimate updated using 1 a Slide state51transition model and measurements. x k|k-1 denotes the Introductionis Section 0 Lecture estimate of the system's state at time step k before the k th measurement yk has been taken into account; Pk|k-1 is the corresponding uncertainty. --Wikipedia, 2013. INTRODUCTION TO Modern Physics PHYX 2710 Ludger Scherliess, of USU Physics, is a world expert at using Kalman filtering for the assimilation Fall 2004 of satellite and ground-based data and the USU GAMES model to predict space weather . Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 51 Intermediate Lab

PHYS 3870 Introduction Rejecting Data Chauvenets Criterion Section 0 Lecture 1 Slide 52 References: Taylor Ch. 6 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 52 Rejecting Data What is a good criteria for rejecting data? 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 Chauvenets criterion provides a (quantitative) compromise

Introduction Section 0 Lecture 1 Slide 53 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 53 Rejecting Data Zallens 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 54 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 54

Rejecting Data Disneys 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 Disneys First Law Wishing will make it so. Introduction Section 0 Lecture 1 Slide 55 Disneys 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 55 Rejecting Data Chauvenets 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 and x 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 56 n = (expected number as deviant as xsuspect) INTRODUCTION TO Modern Physics PHYX 2710 = N Prob(outside xsuspectx) Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 56 Error Function of Gaussian Distribution Error Function: (probability that t

Introduction in Table at right. Section 0 Lecture 1 Slide 57 INTRODUCTION TO Modern Physics PHYX 2710 Probable Error: (probability that Fall 2004 0.67

DISTRIBUTION FUNCTIONS Lecture 3 Slide 58 Chauvenets CriterionExample 1 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 Chauvenets Details (1) 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 Chauvenets CriterionDetails (2) 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 Chauvenets Criterion Details (3) 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 Chauvenets Criterion Example 2 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 Intermediate Lab PHYS 3870

Summary of Probability Theory 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-I 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 Summary of Probability Theory-II Introduction Section 0 Lecture 1 Slide 66 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 66 Summary of Probability Theory-III Introduction Section 0 Lecture 1 Slide 67 INTRODUCTION TO Modern Physics PHYX 2710

Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 67 Summary of Probability Theory-IV Introduction Section 0 Lecture 1 Slide 68 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DISTRIBUTION FUNCTIONS Lecture 3 Slide 68