A Laboratory on Pulse Trains, Counting Statistics and the Central Limit Theorem
|
|
|
- Valentine Watkins
- 5 years ago
- Views:
Transcription
1 A Laboratory on Pulse Trains, Counting Statistics and the Central Limit Theorem David B. Pengra Department of Physics University of Washington Seattle WA 98195
2 The Experiment Cosmic rays hit scintillator paddles and create pulses. PMTs A SCINTILLATOR PADDLES B MUON PATH DISC COINCIDENCE GATE DISC B A A&B AAPT Winter 2006 p.1/2
3 The Experiment Cosmic rays hit scintillator paddles and create pulses. Coincidence used to separate real events from noise. PMTs A B SCINTILLATOR PADDLES Muon Event A pulses MUON PATH Noise DISC B pulses COINCIDENCE GATE DISC B A A&B T A&B coincidence AAPT Winter 2006 p.1/2
4 The Experiment Intervals between successive pulses are captured... Time t t t t t t t T 12 T23 T34 T 45 T 46 T46 Interval lengths AAPT Winter 2006 p.2/2
5 The Experiment... and processed by a LabVIEW application. AAPT Winter 2006 p.3/2
6 Making an interval-statistics theory Basic rules of probability: Rule I. Compound probabilities. If events A and B are mutually exclusive then P (A OR B) = P (A)+P (B). e.g., want 1 or 2 from one die cast. Rule II. Independent probabilities. If events A and B are independent, then P (A AND B) = P (A) P (B). e.g., want snake-eyes (1&1) from two dice cast. AAPT Winter 2006 p.4/2
7 Making an interval-statistics theory Basic rules of probability: Rule I. Compound probabilities. If events A and B are mutually exclusive then P (A OR B) = P (A)+P (B). e.g., want 1 or 2 from one die cast. Rule II. Independent probabilities. If events A and B are independent, then P (A AND B) = P (A) P (B). e.g., want snake-eyes (1&1) from two dice cast. The Fundamental Assumption: Any infinitesimally small interval of time dt is equally likely to contain a pulse. The probability of finding a pulse in dt is given by r dt, where r is a constant, known as the average rate. AAPT Winter 2006 p.4/2
8 Probability of 0 pulses P 0 (T ) is the probability of finding zero pulses in a finite time T. Possible pulses T AAPT Winter 2006 p.5/2
9 Probability of 0 pulses P 0 (T ) is the probability of finding zero pulses in a finite time T. Possible pulses T Τ=Τ/Ν N small intervals AAPT Winter 2006 p.5/2
10 Probability of 0 pulses P 0 (T ) is the probability of finding zero pulses in a finite time T. Possible pulses T Τ=Τ/Ν N small intervals P 0 ( T ) = 1 r T (Rule I: P 0 + P 0 = 1). AAPT Winter 2006 p.5/2
11 Probability of 0 pulses P 0 (T ) is the probability of finding zero pulses in a finite time T. Possible pulses T Τ=Τ/Ν N small intervals P 0 ( T ) = 1 r T (Rule I: P 0 + P 0 = 1). P 0 (T ) = [P 0 ( T )] N = (1 r T ) N (Rule II: independent events). AAPT Winter 2006 p.5/2
12 Probability of 0 pulses P 0 (T ) is the probability of finding zero pulses in a finite time T. Possible pulses T Τ=Τ/Ν N small intervals P 0 ( T ) = 1 r T (Rule I: P 0 + P 0 = 1). P 0 (T ) = [P 0 ( T )] N = (1 r T ) N (Rule II: independent events). Let T = T/N, take limit: ( P 0 (T ) = lim 1 rt N N ) N = e rt AAPT Winter 2006 p.5/2
13 Distribution of 1-pulse intervals over t Let I 1 (t) dt be the differential probability of a pulse interval of length t: i.e., a pulse in t t + dt. 0 t dt NO PULSE PULSE? By Rule II: I 1 (t) dt = ( Probability of no pulse from 0 to t ) ( Probability of one pulse in t t + dt ) AAPT Winter 2006 p.6/2
14 Distribution of 1-pulse intervals over t Let I 1 (t) dt be the differential probability of a pulse interval of length t: i.e., a pulse in t t + dt. 0 t dt NO PULSE PULSE? By Rule II: I 1 (t) dt = ( Probability of no pulse from 0 to t ) ( Probability of one pulse in t t + dt ) = P 0 (t) r dt AAPT Winter 2006 p.6/2
15 Distribution of 1-pulse intervals over t Let I 1 (t) dt be the differential probability of a pulse interval of length t: i.e., a pulse in t t + dt. 0 t dt NO PULSE PULSE? By Rule II: I 1 (t) dt = ( Probability of no pulse from 0 to t ) ( Probability of one pulse in t t + dt ) = P 0 (t) r dt = e rt r dt. So is the interval distribution function. I 1 (t) = re rt AAPT Winter 2006 p.6/2
16 Experimental Results I Interval times collected and histogrammed. Low counts show large fluctuations... AAPT Winter 2006 p.7/2
17 Experimental Results I Interval times collected and histogrammed.... which smooth out at high counts... AAPT Winter 2006 p.7/2
18 Experimental Results I Interval times collected and histogrammed.... and conform to theoretical formula... AAPT Winter 2006 p.7/2
19 Experimental Results I Interval times collected and histogrammed.... which is evident on semilog scale. AAPT Winter 2006 p.7/2
20 How about 2-pulse intervals? Theory: Want probability of an interval that has one pulse somewhere in it. 0 t dt NO PULSE PULSE? From Rule II I 2 (t) dt = ( Probability of no pulse from 0 to t ) ( Probability of one pulse in t to t + dt ) AAPT Winter 2006 p.8/2
21 How about 2-pulse intervals? Theory: Want probability of an interval that has one pulse somewhere in it. 0 t NO PULSE NO PULSE t dt PULSE? PULSE! From Rule II I 2 (t) dt = ( ( Probability of no pulse from 0 to t Probability of no pulse from t to t ) ) ( ( Probability of one pulse in t to t + dt Probability of one pulse in t to t + dt ) ) AAPT Winter 2006 p.8/2
22 How about 2-pulse intervals? Theory: Want probability of an interval that has one pulse somewhere in it. 0 t NO PULSE NO PULSE t dt PULSE? From Rule II and Rule I I 2 (t) dt = all t ( ( PULSE! Probability of no pulse from 0 to t Probability of no pulse from t to t ) ) ( ( Probability of one pulse in t to t + dt Probability of one pulse in t to t + dt ) ) AAPT Winter 2006 p.8/2
23 How about 2-pulse intervals? Theory: Want probability of an interval that has one pulse somewhere in it. 0 t NO PULSE NO PULSE t dt PULSE? From Rule II and Rule I I 2 (t) dt = all t ( = ( ( t 0 PULSE! Probability of no pulse from 0 to t Probability of no pulse from t to t ) ) e rt r e r(t t ) r dt ) ( ( dt Probability of one pulse in t to t + dt Probability of one pulse in t to t + dt ) ) AAPT Winter 2006 p.8/2
24 2-pulse n-pulse intervals. Note structure of calculation: I 2 (t) = t 0 [e t r r] [e (t t )r r] dt = t 0 I 1 (t )I 1 (t t ) dt I 2 (t) is the convolution of two 1-pulse interval functions. AAPT Winter 2006 p.9/2
25 2-pulse n-pulse intervals. Note structure of calculation: I 2 (t) = t 0 [e t r r] [e (t t )r r] dt = t 0 I 1 (t )I 1 (t t ) dt I 2 (t) is the convolution of two 1-pulse interval functions. Similar arguments show I 3 (t) = t 0 I 2 (t )I 1 (t t ) dt AAPT Winter 2006 p.9/2
26 2-pulse n-pulse intervals. Note structure of calculation: I 2 (t) = t 0 [e t r r] [e (t t )r r] dt = t 0 I 1 (t )I 1 (t t ) dt I 2 (t) is the convolution of two 1-pulse interval functions. Similar arguments show I 3 (t) = t 0 I 2 (t )I 1 (t t ) dt And generally I n (t) = t 0 I n 1 (t )I 1 (t t ) dt AAPT Winter 2006 p.9/2
27 A little more math... Iterative elementary calculus shows I 2 (t) = r rt e rt 1 I 3 (t) = r (rt)2 e rt 1 2 And so, (by induction) I 4 (t) = r (rt)3 e rt I n (t) = r (rt)(n 1) e rt (n 1)! AAPT Winter 2006 p.10/2
28 The Erlang distribution This is the integer-gamma or Erlang distribution. 1 Erlang Distribution 1.5 with rescaled t 0.8 n = 1 n= n=5 (1/r)I n (t) n = 2 n = 3 n = 4 n = 5 (n/r)i n (t) 0.5 n=2 n= rt rt/n AAPT Winter 2006 p.11/2
29 Experimental Results II Obtain n-interval distributions by summing 1-interval times, e.g., T 25 = T 23 + T 34 + T 45 for a 3-pulse interval. I 1 (t): t 1 = 0.046s, σ 2 1 = s2 I 1 and theory AAPT Winter 2006 p.12/2
30 Experimental Results II Obtain n-interval distributions by summing 1-interval times, e.g., T 25 = T 23 + T 34 + T 45 for a 3-pulse interval. I 1 (t): t 1 = 0.046s, σ 2 1 = s2 I 2 (t): t 2 = 0.092s, σ 2 2 = s2 I 2 and theory AAPT Winter 2006 p.12/2
31 Experimental Results II Obtain n-interval distributions by summing 1-interval times, e.g., T 25 = T 23 + T 34 + T 45 for a 3-pulse interval. I 1 (t): t 1 = 0.046s, σ 2 1 = s2 I 2 (t): t 2 = 0.092s, σ 2 2 = s2 I 5 (t): t 5 = 0.232s, σ 2 5 = s2 I 5 and theory AAPT Winter 2006 p.12/2
32 Experimental Results II Obtain n-interval distributions by summing 1-interval times, e.g., T 25 = T 23 + T 34 + T 45 for a 3-pulse interval. I 1 (t): t 1 = 0.046s, σ 2 1 = s2 I 2 (t): t 2 = 0.092s, σ 2 2 = s2 I 5 (t): t 5 = 0.232s, σ 2 5 = s2 I 20 (t): t 20 = 0.928s, σ 2 20 = s2 I 20 and theory: becoming normal AAPT Winter 2006 p.12/2
33 Experimental Results II Obtain n-interval distributions by summing 1-interval times, e.g., T 25 = T 23 + T 34 + T 45 for a 3-pulse interval. I 1 (t): t 1 = 0.046s, σ 2 1 = s2 I 2 (t): t 2 = 0.092s, σ 2 2 = s2 I 5 (t): t 5 = 0.232s, σ 2 5 = s2 I 20 (t): t 20 = 0.928s, σ 2 20 = s2 Note: t n nt 1 I 20 and theory: becoming normal σ 2 n nσ 2 1 AAPT Winter 2006 p.12/2
34 The Poisson distribution Another way to look at the result: I n (t) dt = r (rt)(n 1) e rt (n 1)! dt AAPT Winter 2006 p.13/2
35 The Poisson distribution Another way to look at the result: I n (t) dt = r (rt)(n 1) e rt (n 1)! = Probability of n 1 pulses from 0 to t dt ( Probability of one pulse in t to t + dt ) AAPT Winter 2006 p.13/2
36 The Poisson distribution Another way to look at the result: I n (t) dt = r (rt)(n 1) e rt (n 1)! = Probability of n 1 pulses from 0 to t dt ( Probability of one pulse in t to t + dt ) First term is Poisson distribution: P (n; µ) = µn e µ for µ = rt = the average number of counts in fixed t with n being the variable. n! AAPT Winter 2006 p.13/2
37 Experimental Results III Interval-length data set Time t t t t 1 2 t3 t4 5 6 t 7 AAPT Winter 2006 p.14/2
38 Experimental Results III Interval-length data set divided up into fixed time lengths Time t t t t 1 2 t3 t t 7 AAPT Winter 2006 p.14/2
39 Experimental Results III Interval-length data set divided up into fixed time lengths and counts histogrammed. Time t 2 t 2 t3 t4 2 t t t 7 AAPT Winter 2006 p.14/2
40 Experimental Results III For longer counting times mean increases & variance mean. AAPT Winter 2006 p.15/2
41 Comparison of two distributions Poisson (discrete: n) Erlang (continuous: t) mean n = 6.48 mean t = 0.28s σ 2 = 5.60 σ 2 = 0.012s 2 t = 0.3s n = 6 Same formula, different distribution! AAPT Winter 2006 p.16/2
42 Simulations Experiments verify the Poisson process and the exponential 1-interval distribution function are empirical ones. AAPT Winter 2006 p.17/2
43 Simulations Experiments verify the Poisson process and the exponential 1-interval distribution function are empirical ones. Other distributions exist, e.g., 0.4 Uniform 0.4 Normal AAPT Winter 2006 p.17/2
44 Simulations Experiments verify the Poisson process and the exponential 1-interval distribution function are empirical ones. Other distributions exist, e.g., 0.4 Uniform 0.4 Normal which don t follow from Fundamental Assumption. AAPT Winter 2006 p.17/2
45 Simulations Experiments verify the Poisson process and the exponential 1-interval distribution function are empirical ones. Other distributions exist, e.g., 0.4 Uniform 0.4 Normal which don t follow from Fundamental Assumption. These can be simulated. AAPT Winter 2006 p.17/2
46 Simulations I n (t) can be computed for simulated distributions: Empirically: Same method as experiment Manipulate array of 1-interval times. Theoretically: Convolution method applies basic result from statistical theory. I 1 (t) I n (t) AAPT Winter 2006 p.18/2
47 Simulations I n (t) can be computed for simulated distributions: Empirically: Same method as experiment Manipulate array of 1-interval times. Theoretically: Convolution method applies basic result from statistical theory. I 1 (t) I n (t) Exponential Erlang AAPT Winter 2006 p.18/2
48 Simulations I n (t) can be computed for simulated distributions: Empirically: Same method as experiment Manipulate array of 1-interval times. Theoretically: Convolution method applies basic result from statistical theory. I 1 (t) I n (t) Exponential Erlang Uniform Piecewise polynomial AAPT Winter 2006 p.18/2
49 Simulations I n (t) can be computed for simulated distributions: Empirically: Same method as experiment Manipulate array of 1-interval times. Theoretically: Convolution method applies basic result from statistical theory. I 1 (t) I n (t) Exponential Erlang Uniform Piecewise polynomial Normal Normal (Doh!) AAPT Winter 2006 p.18/2
50 Experimental Results IV For Uniform 1-interval distribution (r = 0.1/s, σ 2 = 1/(3r 2 )) I 1 (t): t 1 = 10.1s, σ 2 1 = 33.9s2 I 1 with 1 constant AAPT Winter 2006 p.19/2
51 Experimental Results IV For Uniform 1-interval distribution (r = 0.1/s, σ 2 = 1/(3r 2 )) I 1 (t): t 1 = 10.1s, σ 2 1 = 33.9s2 I 2 (t): t 2 = 20.1s, σ 2 2 = 67.7s2 I 2 with 2 lines AAPT Winter 2006 p.19/2
52 Experimental Results IV For Uniform 1-interval distribution (r = 0.1/s, σ 2 = 1/(3r 2 )) I 1 (t): t 1 = 10.1s, σ 2 1 = 33.9s2 I 2 (t): t 2 = 20.1s, σ 2 2 = 67.7s2 I 3 (t): t 3 = 30.2s, σ 2 3 = 101.6s2 I 3 with 3 parabolas AAPT Winter 2006 p.19/2
53 Experimental Results IV For Uniform 1-interval distribution (r = 0.1/s, σ 2 = 1/(3r 2 )) I 1 (t): t 1 = 10.1s, σ 2 1 = 33.9s2 I 2 (t): t 2 = 20.1s, σ 2 2 = 67.7s2 I 3 (t): t 3 = 30.2s, σ 2 3 = 101.6s2 I 4 (t): t 4 = 40.2s, σ 2 4 = 135.7s2 I 4 with 4 cubics AAPT Winter 2006 p.19/2
54 Experimental Results IV For Uniform 1-interval distribution (r = 0.1/s, σ 2 = 1/(3r 2 )) I 1 (t): t 1 = 10.1s, σ 2 1 = 33.9s2 I 2 (t): t 2 = 20.1s, σ 2 2 = 67.7s2 I 3 (t): t 3 = 30.2s, σ 2 3 = 101.6s2 I 4 (t): t 4 = 40.2s, σ 2 4 = 135.7s2 I 4 with normal (Gaussian) AAPT Winter 2006 p.19/2
55 Experimental Results IV For Uniform 1-interval distribution (r = 0.1/s, σ 2 = 1/(3r 2 )) I 1 (t): t 1 = 10.1s, σ 2 1 = 33.9s2 I 2 (t): t 2 = 20.1s, σ 2 2 = 67.7s2 I 3 (t): t 3 = 30.2s, σ 2 3 = 101.6s2 I 4 (t): t 4 = 40.2s, σ 2 4 = 135.7s2 Again, note: t n nt 1 I 4 with normal (Gaussian) σ 2 n nσ 2 1 AAPT Winter 2006 p.19/2
56 The Central Limit Theorem For all 1-interval distributions, n-interval distribution becomes more symmetrical; AAPT Winter 2006 p.20/2
57 The Central Limit Theorem For all 1-interval distributions, n-interval distribution becomes more symmetrical; shows n-interval mean n 1-interval mean; AAPT Winter 2006 p.20/2
58 The Central Limit Theorem For all 1-interval distributions, n-interval distribution becomes more symmetrical; shows n-interval mean n 1-interval mean; shows n-interval variance n 1-interval variance. AAPT Winter 2006 p.20/2
59 The Central Limit Theorem For all 1-interval distributions, n-interval distribution becomes more symmetrical; shows n-interval mean n 1-interval mean; shows n-interval variance n 1-interval variance. This is the essence of the central limit theorem. AAPT Winter 2006 p.20/2
60 The Central Limit Theorem For all 1-interval distributions, n-interval distribution becomes more symmetrical; shows n-interval mean n 1-interval mean; shows n-interval variance n 1-interval variance. This is the essence of the central limit theorem. Regardless of shape of 1-interval distribution, n-interval distribution tends to normal with well-defined mean and variance. AAPT Winter 2006 p.20/2
61 Concluding remarks Students see the following in this lab: Intervals between random cosmic-ray detection or nuclear decay follows exponential distribution. AAPT Winter 2006 p.21/2
62 Concluding remarks Students see the following in this lab: Intervals between random cosmic-ray detection or nuclear decay follows exponential distribution. Erlang n-interval distribution can be found by direct application of probability rules and Fundamental Assumption. AAPT Winter 2006 p.21/2
63 Concluding remarks Students see the following in this lab: Intervals between random cosmic-ray detection or nuclear decay follows exponential distribution. Erlang n-interval distribution can be found by direct application of probability rules and Fundamental Assumption. Poisson distribution is seen as another way to look at the Erlang formula. It falls out of Erlang distribution calculation. Same formula but different distribution! AAPT Winter 2006 p.21/2
64 Concluding remarks Students see the following in this lab: Intervals between random cosmic-ray detection or nuclear decay follows exponential distribution. Erlang n-interval distribution can be found by direct application of probability rules and Fundamental Assumption. Poisson distribution is seen as another way to look at the Erlang formula. It falls out of Erlang distribution calculation. Same formula but different distribution! Simulations drive home point that exponential distribution is experimental one. AAPT Winter 2006 p.21/2
65 Concluding remarks Students see the following in this lab: Intervals between random cosmic-ray detection or nuclear decay follows exponential distribution. Erlang n-interval distribution can be found by direct application of probability rules and Fundamental Assumption. Poisson distribution is seen as another way to look at the Erlang formula. It falls out of Erlang distribution calculation. Same formula but different distribution! Simulations drive home point that exponential distribution is experimental one. Normal distribution in large-n limit obeyed by all distributions is example of the Central Limit Theorem. AAPT Winter 2006 p.21/2
66 Acknowledgments Prof. Henry Lubatti Prof. James Callis Jason Alferness & UW Physics Department AAPT Winter 2006 p.22/2
Poisson Processes for Neuroscientists
Poisson Processes for Neuroscientists Thibaud Taillefumier This note is an introduction to the key properties of Poisson processes, which are extensively used to simulate spike trains. For being mathematical
Cosmic Ray Counting. I v = cm 2 s 1 str 1,
Cosmic Ray Counting Although we (almost) never think about it, we are bombarded by high-energy ionizing radiation every second of our lives. These high energy particles originate in outer space in the
Chapter 4: An Introduction to Probability and Statistics
Chapter 4: An Introduction to Probability and Statistics 4. Probability The simplest kinds of probabilities to understand are reflected in everyday ideas like these: (i) if you toss a coin, the probability
Week 4: Chap. 3 Statistics of Radioactivity
Week 4: Chap. 3 Statistics of Radioactivity Vacuum Technology General use of Statistical Distributions in Radiation Measurements -- Fluctuations in Number --- distribution function models -- Fluctuations
Brief Review of Probability
Brief Review of Probability Nuno Vasconcelos (Ken Kreutz-Delgado) ECE Department, UCSD Probability Probability theory is a mathematical language to deal with processes or experiments that are non-deterministic
ESSENTIALS OF LEARNING. Math 7. Math A MATH B. Pre-Calculus. Math 12X. Visual Basic
Three Viillllage Centtrall Schooll Diisttriictt ESSENTIALS OF LEARNING MATHEMATICS Math 7 Math A MATH B Pre-Calculus Math 12X Visual Basic The mission of the Three Village Central School District, in concert
MEASURING THE LIFETIME OF THE MUON
B6-1 MEASURING THE LIFETIME OF THE MUON Last Revised September 19, 2006 QUESTION TO BE INVESTIGATED What is the lifetime τ of a muon? INTRODUCTION AND THEORY Muons are a member of a group of particles
Cosmic Ray Muon Detection Measurements of Cosmic Ray Muon Flux and Muon Lifetime Using Scintillation Detectors
Cosmic Ray Muon Detection Measurements of Cosmic Ray Muon Flux and Muon Lifetime Using Scintillation Detectors Department of Physics and Space Sciences Florida Institute of Technology G. Karagiorgi, J.
Chapter 8: An Introduction to Probability and Statistics
Course S3, 200 07 Chapter 8: An Introduction to Probability and Statistics This material is covered in the book: Erwin Kreyszig, Advanced Engineering Mathematics (9th edition) Chapter 24 (not including
Math 353 Lecture Notes Week 6 Laplace Transform: Fundamentals
Math 353 Lecture Notes Week 6 Laplace Transform: Fundamentals J. Wong (Fall 217) October 7, 217 What did we cover this week? Introduction to the Laplace transform Basic theory Domain and range of L Key
Investigation of Combined Positive and Negative Muon Decay in a Scintillator
Investigation of Combined Positive and Negative Muon Decay in a Scintillator Brian Lim and D. Ruben Cornell University, Ithaca, NY 14853, USA (received 7 November 005) Based on some classic experiments
Waveform Analysis for DM-Ice17. Zachary Pierpoint University of Wisconsin - Madison October 21, 2013 Yale Weak Interactions Discussions Group
Waveform Analysis for DM-Ice17 Zachary Pierpoint University of Wisconsin - Madison October 21, 213 Yale Weak Interactions Discussions Group DM-Ice17 Energy Spectrum counts / day / kev / kg 1 28 Tl+ 214
HiSPARC Detector - Detector Station
HiSPARC Detector - Detector Station Koos Kortland translated and adapted by K. Schadenberg 1 Introduction This module is a part of a series describing the HiSPARC detector. A detector station consists
Introductory Mathematics
Introductory Mathematics 1998 2003 1.01 Identify subsets of the real number system. 1.02 Estimate and compute with rational Grade 7: 1.02 numbers. 1.03 Compare, order, and convert among Grade 6: 1.03 fractions,
Binomials defined, 13 division by, FOIL method and, 22 multiplying binomial by trinomial,
5639_Holtfrerich_Index 6/2/05 11:45 AM Page I-1 Index Absolute value defined, 46 functions and, 126 59 Absolute value equations, solutions, 46 49 Absolute value inequalities, solutions, 263 267 Acceleration,
Measuring Cosmic Ray Muon Decay Constant and Flux
WJP, PHY381 (2015) Wabash Journal of Physics v3.3, p.1 Measuring Cosmic Ray Muon Decay Constant and Flux R.C. Dennis, D.T. Tran, J. Qi, and J. Brown Department of Physics, Wabash College, Crawfordsville,
Chapter 2 Random Variables
Stochastic Processes Chapter 2 Random Variables Prof. Jernan Juang Dept. of Engineering Science National Cheng Kung University Prof. Chun-Hung Liu Dept. of Electrical and Computer Eng. National Chiao Tung
Algebra II- Comprehensive/ Part A
Algebra II- Comprehensive/ Part A COURSE DESCRIPTION: This course builds upon algebraic concepts covered in Algebra I and prepares students for advanced-level courses. Students extend their knowledge and
CHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE ALGEBRA II
CHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE ALGEBRA II Course Number 5116 Department Mathematics Qualification Guidelines Successful completion of both semesters of Algebra 1 or Algebra 1
CS 246 Review of Proof Techniques and Probability 01/14/19
Note: This document has been adapted from a similar review session for CS224W (Autumn 2018). It was originally compiled by Jessica Su, with minor edits by Jayadev Bhaskaran. 1 Proof techniques Here we
Calibration of the Modular Neutron Array (MoNA)
Calibration of the Modular Neutron Array (MoNA) Robert Pepin Gonzaga University ~1~ Calibration of the Modular Neutron Array (MoNA): Figure 1 - A rendering of the Modular Neutron Array In recent years
Physics 248, Spring 2009 Lab 6: Radiation and its Interaction with Matter
Name Section Physics 48, Spring 009 Lab 6: Radiation and its Interaction with Matter Your TA will use this sheet to score your lab. It is to be turned in at the end of lab. To receive full credit you must
5.1 Polynomial Functions
5.1 Polynomial Functions In this section, we will study the following topics: Identifying polynomial functions and their degree Determining end behavior of polynomial graphs Finding real zeros of polynomial
Rice University Physics 332 LIFETIME OF THE MUON I. INTRODUCTION...2! II. MEASUREMENT PROCEDURES...3! III. ANALYSIS PROCEDURES...7!
Rice University Physics 332 LIFETIME OF THE MUON I. INTRODUCTION...2! II. MEAUREMENT PROCEDURE...3! III. ANALYI PROCEDURE...7! Revised July 2011 I. Introduction In this experiment you will measure the
Solutions. Some of the problems that might be encountered in collecting data on check-in times are:
Solutions Chapter 7 E7.1 Some of the problems that might be encountered in collecting data on check-in times are: Need to collect separate data for each airline (time and cost). Need to collect data for
Flux Studies: Muons & Gamma Rays. By: Akkshay Khoslaa & Duncan Wilmot
Flux Studies: Muons & Gamma Rays By: Akkshay Khoslaa & Duncan Wilmot Calibration Graphs 50 45 40 35 30 25 20 15 10 5 0 0.5 0.6 0.7 0.8 0.9 25 20 15 10 5 0 0.55 0.65 0.75 0.85 0.95 1.05 A & B Coincident
IE 230 Probability & Statistics in Engineering I. Closed book and notes. 60 minutes.
Closed book and notes. 60 minutes. A summary table of some univariate continuous distributions is provided. Four Pages. In this version of the Key, I try to be more complete than necessary to receive full
LAB 2 1. Measurement of 2. Binomial Distribution
LAB 2 Gan Phys 3700 1. Measurement of π In this exercise we will determine a value for π by throwing darts: a) Determine π by throwing a dart 100 or more times. Use an 8x11 inch sheet of paper with a circle
Statistics (1): Estimation
Statistics (1): Estimation Marco Banterlé, Christian Robert and Judith Rousseau Practicals 2014-2015 L3, MIDO, Université Paris Dauphine 1 Table des matières 1 Random variables, probability, expectation
Cosmic Muon Shower Study with QuarkNet
WJP, PHY381 (2015) Wabash Journal of Physics v4.2, p.1 Cosmic Muon Shower Study with QuarkNet Brian Hayhurst, Jia Qi, and James Brown Department of Physics, Wabash College, Crawfordsville, IN 47933 (Dated:
Computer Vision & Digital Image Processing
Computer Vision & Digital Image Processing Image Restoration and Reconstruction I Dr. D. J. Jackson Lecture 11-1 Image restoration Restoration is an objective process that attempts to recover an image
Poisson Variables. Robert DeSerio
Poisson Variables Robert DeSerio Poisson processes A Poisson process is one in which events are randomly distributed in time, space or some other variable with the number of events in any non-overlapping
2.3 Estimating PDFs and PDF Parameters
.3 Estimating PDFs and PDF Parameters estimating means - discrete and continuous estimating variance using a known mean estimating variance with an estimated mean estimating a discrete pdf estimating a
PHYS 114 Exam 1 Answer Key NAME:
PHYS 4 Exam Answer Key AME: Please answer all of the questions below. Each part of each question is worth points, except question 5, which is worth 0 points.. Explain what the following MatLAB commands
Cosmic Ray Muons Research Project
Cosmic Ray Muons Research Project School of Physics and Astronomy, Queen Mary University of London Abstract Muons are fundamental particles similar to electrons which can be created by cosmic rays bombarding
The degree of the polynomial function is n. We call the term the leading term, and is called the leading coefficient. 0 =
Math 1310 A polynomial function is a function of the form = + + +...+ + where 0,,,, are real numbers and n is a whole number. The degree of the polynomial function is n. We call the term the leading term,
AQA Statistics 1. Probability. Section 1: Introducing Probability. Notation
Notes and Examples AQA Statistics 1 Probability Section 1: Introducing Probability These notes contain subsections on Notation The complement of an event Probability of either one event or another Notation
arxiv: v1 [physics.ins-det] 3 Feb 2011
![arxiv: v1 [physics.ins-det] 3 Feb 2011](/thumbs/86/94571492.jpg "arxiv: v1 [physics.ins-det] 3 Feb 2011")
Nuclear Instruments and Methods in Physics Research A 00 (2018) 1 5 Alogo.pdf Nuclear Instruments and Methods in Physics Research A Scintillation decay time and pulse shape discrimination in oxygenated
Probability, For the Enthusiastic Beginner (Exercises, Version 1, September 2016) David Morin,
Chapter 8 Exercises Probability, For the Enthusiastic Beginner (Exercises, Version 1, September 2016) David Morin, morin@physics.harvard.edu 8.1 Chapter 1 Section 1.2: Permutations 1. Assigning seats *
The Geometry of Cubic Maps
The Geometry of Cubic Maps John Milnor Stony Brook University (www.math.sunysb.edu) work with Araceli Bonifant and Jan Kiwi Conformal Dynamics and Hyperbolic Geometry CUNY Graduate Center, October 23,
Grade Math (HL) Curriculum
Grade 11-12 Math (HL) Curriculum Unit of Study (Core Topic 1 of 7): Algebra Sequences and Series Exponents and Logarithms Counting Principles Binomial Theorem Mathematical Induction Complex Numbers Uses
PRECALCULUS BISHOP KELLY HIGH SCHOOL BOISE, IDAHO. Prepared by Kristina L. Gazdik. March 2005
PRECALCULUS BISHOP KELLY HIGH SCHOOL BOISE, IDAHO Prepared by Kristina L. Gazdik March 2005 1 TABLE OF CONTENTS Course Description.3 Scope and Sequence 4 Content Outlines UNIT I: FUNCTIONS AND THEIR GRAPHS
Finish up our overview of small and large
Finish up our overview of small and large Lecture 5 Limits of our knowledge Clicker practice quiz Some terminology... "Elementary particles" = objects that make up atoms (n,p,e) or are produced when atoms
How do we analyze, evaluate, solve, and graph quadratic functions?
Topic: 4. Quadratic Functions and Factoring Days: 18 Key Learning: Students will be able to analyze, evaluate, solve and graph quadratic functions. Unit Essential Question(s): How do we analyze, evaluate,
Common probability distributionsi Math 217 Probability and Statistics Prof. D. Joyce, Fall 2014
Introduction. ommon probability distributionsi Math 7 Probability and Statistics Prof. D. Joyce, Fall 04 I summarize here some of the more common distributions used in probability and statistics. Some
MATH 324 Summer 2011 Elementary Number Theory. Notes on Mathematical Induction. Recall the following axiom for the set of integers.
MATH 4 Summer 011 Elementary Number Theory Notes on Mathematical Induction Principle of Mathematical Induction Recall the following axiom for the set of integers. Well-Ordering Axiom for the Integers If
W. Udo Schröder Departments of Chemistry & of Physics and Astronomy
W. Udo Schröder Departments of Chemistry & of Physics and Astronomy ANSEL Faculty Instructors ACS NuSci Acad Infrastructure 2 Prof. Frank Wolfs Prof. Udo Schrőder Research: Large Underground Xenon (LUX)
STATE COUNCIL OF EDUCATIONAL RESEARCH AND TRAINING TNCF DRAFT SYLLABUS.
STATE COUNCIL OF EDUCATIONAL RESEARCH AND TRAINING TNCF 2017 - DRAFT SYLLABUS Subject :Mathematics Class : XI TOPIC CONTENT Unit 1 : Real Numbers - Revision : Rational, Irrational Numbers, Basic Algebra
Quantitative Methods Chapter 0: Review of Basic Concepts 0.1 Business Applications (II) 0.2 Business Applications (III)
Quantitative Methods Chapter 0: Review of Basic Concepts 0.1 Business Applications (II) 0.1.1 Simple Interest 0.2 Business Applications (III) 0.2.1 Expenses Involved in Buying a Car 0.2.2 Expenses Involved
Latest Syllabus - NMO
Latest Syllabus - NMO CLASS 1 Numerals and Number Name (2 digit numbers) Addition, Subtraction Knowledge of currency notes. Knowledge of clock. Knowledge of basic units of Length,. Knowledge of basic figures
PRACTICE PROBLEMS FOR EXAM 2
PRACTICE PROBLEMS FOR EXAM 2 Math 3160Q Fall 2015 Professor Hohn Below is a list of practice questions for Exam 2. Any quiz, homework, or example problem has a chance of being on the exam. For more practice,
Synthesis of plastic scintillator. Ildefonso León Monzón Universidad Autónoma de Sinaloa
Synthesis of plastic scintillator Ildefonso León Monzón Universidad Autónoma de Sinaloa MUON BUNDLE triggered by plastic scintillator modules from ACORDE Plastic Scintillator how does it work? Excitation
( ) is symmetric about the y - axis.
(Section 1.5: Analyzing Graphs of Functions) 1.51 PART F: FUNCTIONS THAT ARE EVEN / ODD / NEITHER; SYMMETRY A function f is even f ( x) = f ( x) x Dom( f ) The graph of y = f x for every x in the domain
Portable Assisted Study Sequence ALGEBRA IIB
SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The second half of
Sample Spectroscopy System Hardware
Semiconductor Detectors vs. Scintillator+PMT Detectors Semiconductors are emerging technology - Scint.PMT systems relatively unchanged in 50 years. NaI(Tl) excellent for single-photon, new scintillation
Integrated Algebra High School
Niagara Falls City School District 607 Walnut Avenue, Niagara Falls, NY 14301 Integrated Algebra High School Day NYS Performance Indicators Objectives 1 Pre-Test 2 MST3.IA.AL7.01 MST3.IA.AL7.04 Chapter
SMSU Mathematics Course Content
Southwest Minnesota State University Department of Mathematics SMSU Mathematics Course Content 2012-2013 Thefollowing is a list of possibletopics and techniques to cover in your SMSU College Now course.
Topic Subtopics Essential Knowledge (EK)
Unit/ Unit 1 Limits [BEAN] 1.1 Limits Graphically Define a limit (y value a function approaches) One sided limits. Easy if it s continuous. Tricky if there s a discontinuity. EK 1.1A1: Given a function,
MTH306: Algebra II. Course length: Two semesters. Materials: None
MTH306: Algebra II Students are able to gain credit if they have previously completed this course but did not successfully earn credit. For each unit, students take a diagnostic test that assesses their
Parametric Techniques Lecture 3
Parametric Techniques Lecture 3 Jason Corso SUNY at Buffalo 22 January 2009 J. Corso (SUNY at Buffalo) Parametric Techniques Lecture 3 22 January 2009 1 / 39 Introduction In Lecture 2, we learned how to
EE/CpE 345. Modeling and Simulation. Fall Class 5 September 30, 2002
EE/CpE 345 Modeling and Simulation Class 5 September 30, 2002 Statistical Models in Simulation Real World phenomena of interest Sample phenomena select distribution Probabilistic, not deterministic Model
Chapter Learning Objectives. Probability Distributions and Probability Density Functions. Continuous Random Variables
Chapter 4: Continuous Random Variables and Probability s 4-1 Continuous Random Variables 4-2 Probability s and Probability Density Functions 4-3 Cumulative Functions 4-4 Mean and Variance of a Continuous
Probability Distributions - Lecture 5
Probability Distributions - Lecture 5 1 Introduction There are a number of mathematical models of probability density functions that represent the behavior of physical systems. In this lecture we explore
Measuring the Muon Lifetime
WJP, PHY38 (200) Wabash Journal of Physics v4.0, p. Measuring the Muon Lifetime L.W. Lupinski, R. Paudel, and M.J. Madsen Department of Physics, Wabash College, Crawfordsville, IN 47933 (Dated: March,
MESUREMENT OF MUON LIFETIME USING COSMIC MUON STOPPED IN PLASTIC SCINTILLATOR DETECTOR.
MESUREMENT OF MUON LIFETIME USING COSMIC MUON STOPPED IN PLASTIC SCINTILLATOR DETECTOR. ASMITA REDIJ SCIENTIFIC OFFICER. AS A PART OF EXPERIMENTAL PHYSICS COURSE (I). CONDUCTED FOR INO TRAINING SCHOOL.
1 Probability and Random Variables
1 Probability and Random Variables The models that you have seen thus far are deterministic models. For any time t, there is a unique solution X(t). On the other hand, stochastic models will result in
Parametric Techniques
Parametric Techniques Jason J. Corso SUNY at Buffalo J. Corso (SUNY at Buffalo) Parametric Techniques 1 / 39 Introduction When covering Bayesian Decision Theory, we assumed the full probabilistic structure
Algebra 1 Correlation of the ALEKS course Algebra 1 to the Washington Algebra 1 Standards
Algebra 1 Correlation of the ALEKS course Algebra 1 to the Washington Algebra 1 Standards A1.1: Core Content: Solving Problems A1.1.A: Select and justify functions and equations to model and solve problems.
Region 16 Board of Education. Precalculus Curriculum
Region 16 Board of Education Precalculus Curriculum 2008 1 Course Description This course offers students an opportunity to explore a variety of concepts designed to prepare them to go on to study calculus.
) = nlog b ( m) ( m) log b ( ) ( ) = log a b ( ) Algebra 2 (1) Semester 2. Exponents and Logarithmic Functions
Exponents and Logarithmic Functions Algebra 2 (1) Semester 2! a. Graph exponential growth functions!!!!!! [7.1]!! - y = ab x for b > 0!! - y = ab x h + k for b > 0!! - exponential growth models:! y = a(
PHYSICS 2150 LABORATORY
PHYSICS 2150 LABORATORY Professor John Cumalat TAs: Adam Green John Houlton Lab Coordinator: Scott Pinegar Lecture 6 Feb. 17, 2015 ANNOUNCEMENT The problem set will be posted on the course website or you
Phys 243 Lab 7: Radioactive Half-life
Phys 243 Lab 7: Radioactive Half-life Dr. Robert MacDonald The King s University College Winter 2013 Abstract In today s lab you ll be measuring the half-life of barium-137, a radioactive isotope of barium.
Math 1310 Section 4.1: Polynomial Functions and Their Graphs. A polynomial function is a function of the form ...
Math 1310 Section 4.1: Polynomial Functions and Their Graphs A polynomial function is a function of the form... where 0,,,, are real numbers and n is a whole number. The degree of the polynomial function
Design, Construction, Operation, and Simulation of a Radioactivity Assay Chamber
Design, Construction, Operation, and Simulation of a Radioactivity Assay Chamber Wesley Ketchum and Abe Reddy EWI Group, UW REU 2006 Outline Neutrino Physics Background Double Beta Decay and the Majorana
Neutrinos. Why measure them? Why are they difficult to observe?
Outline What is a neutrino? Why do we want to study them? Building a detector to detect the undetectable What does a neutrino detector see? How do you seperate a neutrino signal from the background? Neutrinos
AP Calculus AB Course Outline
AP Calculus AB Course Outline Prerequisite: Satisfactory completion of: Geometry, Algebra II, and Pre-calculus Advanced Placement Calculus AB is designed as college-level Calculus I. Students are required
Homework # , Spring Due 14 May Convergence of the empirical CDF, uniform samples
Homework #3 36-754, Spring 27 Due 14 May 27 1 Convergence of the empirical CDF, uniform samples In this problem and the next, X i are IID samples on the real line, with cumulative distribution function
Development of a Pulse Shape Discrimination IC
Development of a Pulse Shape Discrimination IC Michael Hall Southern Illinois University Edwardsville VLSI Design Research Laboratory October 20, 2006 Design Team Southern Illinois University Edwardsville:
Radioactivity. Radioactivity
The Law of Radioactive Decay. 72 The law of radioactive decay. It turns out that the probability per unit time for any radioactive nucleus to decay is a constant, called the decay constant, lambda, ".
Random Variables and Their Distributions
Chapter 3 Random Variables and Their Distributions A random variable (r.v.) is a function that assigns one and only one numerical value to each simple event in an experiment. We will denote r.vs by capital
NEW YORK ALGEBRA TABLE OF CONTENTS
NEW YORK ALGEBRA TABLE OF CONTENTS CHAPTER 1 NUMBER SENSE & OPERATIONS TOPIC A: Number Theory: Properties of Real Numbers {A.N.1} PART 1: Closure...1 PART 2: Commutative Property...2 PART 3: Associative
Cumulative Test. 141 Holt Algebra 2. Name Date Class. Select the best answer. 1. Identify the property demonstrated by. 7.
Select the best answer. 1. Identify the property demonstrated by 4 5 6 5 6 4. A Associative Property B Commutative Property C Distributive Property D Additive Identity Property. Simplify 18 5 50 1. F 0
Introduction to Probability and Sample Spaces
2.2 2.3 Introduction to Probability and Sample Spaces Prof. Tesler Math 186 Winter 2019 Prof. Tesler Ch. 2.3-2.4 Intro to Probability Math 186 / Winter 2019 1 / 26 Course overview Probability: Determine
Summary of basic probability theory Math 218, Mathematical Statistics D Joyce, Spring 2016
8. For any two events E and F, P (E) = P (E F ) + P (E F c ). Summary of basic probability theory Math 218, Mathematical Statistics D Joyce, Spring 2016 Sample space. A sample space consists of a underlying
Algebra 2A Unit 1 Week 1 Day Activity Unit 1 Week 2 Day Activity Unit 1 Week 3 Day Activity Unit 2 Week 1 Day Activity
Algebra 2A Unit 1 Week 1 1 Pretest Unit 1 2 Evaluating Rational Expressions 3 Restrictions on Rational Expressions 4 Equivalent Forms of Rational Expressions 5 Simplifying Rational Expressions Unit 1 Week
Chapter 1 Statistical Reasoning Why statistics? Section 1.1 Basics of Probability Theory
Chapter 1 Statistical Reasoning Why statistics? Uncertainty of nature (weather, earth movement, etc. ) Uncertainty in observation/sampling/measurement Variability of human operation/error imperfection
Irr. Statistical Methods in Experimental Physics. 2nd Edition. Frederick James. World Scientific. CERN, Switzerland
Frederick James CERN, Switzerland Statistical Methods in Experimental Physics 2nd Edition r i Irr 1- r ri Ibn World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI CONTENTS
Statistical Methods in Particle Physics
Statistical Methods in Particle Physics Lecture 3 October 29, 2012 Silvia Masciocchi, GSI Darmstadt s.masciocchi@gsi.de Winter Semester 2012 / 13 Outline Reminder: Probability density function Cumulative
Algebra 2 Secondary Mathematics Instructional Guide
Algebra 2 Secondary Mathematics Instructional Guide 2009-2010 ALGEBRA 2AB (Grade 9, 10 or 11) Prerequisite: Algebra 1AB or Geometry AB 310303 Algebra 2A 310304 Algebra 2B COURSE DESCRIPTION Los Angeles
ROSLYN PUBLIC SCHOOLS INTEGRATED ALGEBRA CURRICULUM. Day(s) Topic Textbook Workbook Additional Worksheets
I. Review 1 and 2 Review p3-19 1. Screening Test 2. Review sheet II. Real Numbers A.N.1 Identify and apply the properties of real numbers (closure, commutative, associative, distributive, identity, inverse)
Numerical Analysis: Interpolation Part 1
Numerical Analysis: Interpolation Part 1 Computer Science, Ben-Gurion University (slides based mostly on Prof. Ben-Shahar s notes) 2018/2019, Fall Semester BGU CS Interpolation (ver. 1.00) AY 2018/2019,
MATH-1420 Review Concepts (Haugen)
MATH-40 Review Concepts (Haugen) Unit : Equations, Inequalities, Functions, and Graphs Rational Expressions Determine the domain of a rational expression Simplify rational expressions -factor and then
Measurement of Muon Lifetime
Measurement of Muon Lifetime Noah Scandrette Physics and Astronomy Department, San Francisco State University, San Francisco, California (Dated: December 16, 2016) The average lifetime of the muon has
Single Maths B: Introduction to Probability
Single Maths B: Introduction to Probability Overview Lecturer Email Office Homework Webpage Dr Jonathan Cumming j.a.cumming@durham.ac.uk CM233 None! http://maths.dur.ac.uk/stats/people/jac/singleb/ 1 Introduction
G.PULLAIAH COLLEGE OF ENGINEERING & TECHNOLOGY DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING PROBABILITY THEORY & STOCHASTIC PROCESSES
G.PULLAIAH COLLEGE OF ENGINEERING & TECHNOLOGY DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING PROBABILITY THEORY & STOCHASTIC PROCESSES LECTURE NOTES ON PTSP (15A04304) B.TECH ECE II YEAR I SEMESTER
Neutrino Physics. Neutron Detector in the Aberdeen Tunnel Underground Laboratory. The Daya Bay Experiment. Significance of θ 13
Neutrino Physics Neutron Detector in the Aberdeen Tunnel Underground Laboratory John K.C. Leung Department of Physics, HKU On behalf of CUHK, LBL & IHEP Presented to HKRPS on June 21, 2007 Neutrinos have
Probability Density Functions
Statistical Methods in Particle Physics / WS 13 Lecture II Probability Density Functions Niklaus Berger Physics Institute, University of Heidelberg Recap of Lecture I: Kolmogorov Axioms Ingredients: Set
Introduction and Overview STAT 421, SP Course Instructor
Introduction and Overview STAT 421, SP 212 Prof. Prem K. Goel Mon, Wed, Fri 3:3PM 4:48PM Postle Hall 118 Course Instructor Prof. Goel, Prem E mail: goel.1@osu.edu Office: CH 24C (Cockins Hall) Phone: 614
Modeling Data with Linear Combinations of Basis Functions. Read Chapter 3 in the text by Bishop
Modeling Data with Linear Combinations of Basis Functions Read Chapter 3 in the text by Bishop A Type of Supervised Learning Problem We want to model data (x 1, t 1 ),..., (x N, t N ), where x i is a vector
Algebra II Assessment. Eligible Texas Essential Knowledge and Skills
Algebra II Assessment Eligible Texas Essential Knowledge and Skills STAAR Algebra II Assessment Mathematical Process Standards These student expectations will not be listed under a separate reporting category.