A Laboratory on Pulse Trains, Counting Statistics and the Central Limit Theorem David B. Pengra Department of Physics University of Washington Seattle WA 98195
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
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
The Experiment Intervals between successive pulses are captured... Time t t t t t t t 1 2 3 4 5 6 7 T 12 T23 T34 T 45 T 46 T46 Interval lengths AAPT Winter 2006 p.2/2
The Experiment... and processed by a LabVIEW application. AAPT Winter 2006 p.3/2
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
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
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
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
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
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
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
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
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
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
Experimental Results I Interval times collected and histogrammed. Low counts show large fluctuations... AAPT Winter 2006 p.7/2
Experimental Results I Interval times collected and histogrammed.... which smooth out at high counts... AAPT Winter 2006 p.7/2
Experimental Results I Interval times collected and histogrammed.... and conform to theoretical formula... AAPT Winter 2006 p.7/2
Experimental Results I Interval times collected and histogrammed.... which is evident on semilog scale. AAPT Winter 2006 p.7/2
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
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
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
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
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
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
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
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 1 2 3 I n (t) = r (rt)(n 1) e rt (n 1)! AAPT Winter 2006 p.10/2
The Erlang distribution This is the integer-gamma or Erlang distribution. 1 Erlang Distribution 1.5 with rescaled t 0.8 n = 1 n=10 0.6 1 n=5 (1/r)I n (t) 0.4 0.2 n = 2 n = 3 n = 4 n = 5 (n/r)i n (t) 0.5 n=2 n=1 0 0 1 2 3 4 5 6 7 8 rt 0 0 1 2 3 4 rt/n AAPT Winter 2006 p.11/2
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 = 0.0019s2 I 1 and theory AAPT Winter 2006 p.12/2
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 = 0.0019s2 I 2 (t): t 2 = 0.092s, σ 2 2 = 0.0039s2 I 2 and theory AAPT Winter 2006 p.12/2
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 = 0.0019s2 I 2 (t): t 2 = 0.092s, σ 2 2 = 0.0039s2 I 5 (t): t 5 = 0.232s, σ 2 5 = 0.0097s2 I 5 and theory AAPT Winter 2006 p.12/2
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 = 0.0019s2 I 2 (t): t 2 = 0.092s, σ 2 2 = 0.0039s2 I 5 (t): t 5 = 0.232s, σ 2 5 = 0.0097s2 I 20 (t): t 20 = 0.928s, σ 2 20 = 0.0365s2 I 20 and theory: becoming normal AAPT Winter 2006 p.12/2
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 = 0.0019s2 I 2 (t): t 2 = 0.092s, σ 2 2 = 0.0039s2 I 5 (t): t 5 = 0.232s, σ 2 5 = 0.0097s2 I 20 (t): t 20 = 0.928s, σ 2 20 = 0.0365s2 Note: t n nt 1 I 20 and theory: becoming normal σ 2 n nσ 2 1 AAPT Winter 2006 p.12/2
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
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
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
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
Experimental Results III Interval-length data set divided up into fixed time lengths Time t t t t 1 2 t3 t4 5 6 2 2 3 t 7 AAPT Winter 2006 p.14/2
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 3 1 5 6 t 7 AAPT Winter 2006 p.14/2
Experimental Results III For longer counting times mean increases & variance mean. AAPT Winter 2006 p.15/2
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
Simulations Experiments verify the Poisson process and the exponential 1-interval distribution function are empirical ones. AAPT Winter 2006 p.17/2
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 0.2 0.2 0 0 2 4 6 8 0 0 2 4 6 8 AAPT Winter 2006 p.17/2
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 0.2 0.2 0 0 2 4 6 8 0 0 2 4 6 8... which don t follow from Fundamental Assumption. AAPT Winter 2006 p.17/2
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 0.2 0.2 0 0 2 4 6 8 0 0 2 4 6 8... which don t follow from Fundamental Assumption. These can be simulated. AAPT Winter 2006 p.17/2
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
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
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
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
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
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
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
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
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
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
The Central Limit Theorem For all 1-interval distributions, n-interval distribution becomes more symmetrical; AAPT Winter 2006 p.20/2
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
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
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
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
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
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
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
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
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
Acknowledgments Prof. Henry Lubatti Prof. James Callis Jason Alferness & UW Physics Department AAPT Winter 2006 p.22/2