Basic Counting Statistics

1. Who's Counting? Basic Counting Statistics Basic Counting Statistics

2. 2 sources of stochastic observables x in nuclear science: 1) Nuclear phenomena are governed by quantal wave functions and inherent statistics 2) Detection of processes occurs with imperfect efficiency (e < 1) and finite resolution distributing sharp events x0 over a range in x. Stochastic observables x have a range of values with frequencies determined by probability distribution P(x).characterize by set of moments of P <xn> = ∫ xnP (x)dx; n 0, 1, 2,… Normalization <x0> = 1. First moment (expectation value) of P: E(x) = <x> = ∫xP(x)dx second central moment = “variance” of P(x): sx2 = <x2- <x2> > Stochastic Nuclear Observables Statistics

3. Nuclear systems: quantal wave functions yi(x,…;t) (x,…;t) = degrees of freedom of system, time. Probability density (e.g., for x, integrate over other d.o.f.) Uncertainty and Statistics 1 2 Partial probability rates l12for disappearance (decay of 12) can vary over many orders of magnitude  no certainty  statistics

4. The Normal Distribution 2sx P(x) <x> x P(x) x Continuous function or discrete distribution (over bins) Normalized probability Statistics

5. Experimental Mean Counts and Variance Measured by ensemble sampling expectation values + uncertainties Sample (Ensemble) = Population instant236U(0.25mg) source, count a particles emitted during N = 10 time intervals (samples @1 min). l =?? Statistics Slightly different from sample to sample

6. Sample Statistics P(x) x <x> =4.96 s =0.94 <x> =4.96 s =1.23 <x> =5.11 s =1.11 <x>+s <x> Assume true population distribution for variable x with true (“population”) mean <x>pop = 5.0, nx =1.0 3 independent sample measurements (equivalent statistics): x <x>-s Statistics Mean arithmetic sample average <x> = (5.11+4.96+4.96)/3 = 5.01 Variance of sample averagess2=s2= [(5.11-5.01)2+2(4.96-5.01)2]/2 = 0.01 s= 0.0075 sx2= 0.0075/3 = 0.0025 sx = 0.05 Result: <x>pop 5.01 ± 0.05

7. Example of Gaussian Population Sample size makes a difference ( weighted average) n = 10 n = 50 The larger the sample, the narrower the distribution of x values, the more it approaches the true Gaussian (normal) distribution. Statistics xm xm

8. Increasing size n of samples: Distribution of sample means Gaussian normal distrib. regardless of form of original (population) distribution. Central-Limit Theorem The means (averages) of different samples in the previous example cluster together closely.  general property:  The average of a distribution does not contain information on the shape of the distribution. The average of any truly random sample of a population is already somewhat close to the true population average. Many or large samples narrow the choices: smaller Gaussian width  Standard error of the mean decreases with incr. sample size Statistics

9. Binomial Distribution Integer random variable m = number of events, out of N total, of a given type, e.g., decay of m (from a sample of N )radioactive nuclei, or detection of m (out of N ) photons arriving at detector. p = probability for a (one) success (decay of one nucleus, detection of one photon) Choose an arbitrary sample of m trialsout of N trials pm= probability for at leastm successes (1-p)N-m = probability for N-m failures (survivals, escaping detection) Probability for exactly m successes out of a total of N trials How many ways can m events be chosen out of N ?  Binomial coefficient Total probability (success rate) for any sample of m events: Statistics

10. Moments and Limits Probability for m “successes” out of N trials, individual probability p Distributions for N=30 and p=0.1  p=0.3Poisson  Gaussian Statistics

11. Poisson Probability Distribution Probability for observing m events when average is <m> = m Results from binomial distribution in the limit of small p and large N (N·p > 0) m=0,1,2,… m = <m> = N·pands2 = m is the mean, the average number of successes in N trials. Observe Ncounts (events)  uncertainty iss= √m Unlike the binomial distribution, the Poisson distribution does not depend explicitly on p or N ! For large N, p: Poisson  Gaussian (Normal Distribution) Statistics

12. Moments of Transition Probabilities Small probability for process, but many trials (n0 = 6.38·1017) 0< n0·l < ∞ Statistical process follows a Poisson distribution: n=“random” Different statistical distributions: Binomial, Poisson, Gaussian Statistics

13. Slow radioactive decay of large sample Sample size N » 1, decay probability p « 1, with 0 < N·p <  137Cs unstableisotope  decay t1/2 = 27 years  p = ln2/27 = 0.026/a = 8.2·10-10s-1  0 Sample of 1 mg: N = 1015 nuclei (=trials for decay) How many will decay(= activity m) ? m=< >= N·p = 8.2·10+5 s-1 Count rate estimate < >=d<N>/dt = (8.2·10+5 ± 905)s-1 estimatedProbability for m actual decays P (m,m) = Radioactive Decay as Poisson Process Statistics

14. Random independentvariable sets {N1}, {N2},….,{Nn} corresponding variances s12, s22,….,sn2 Function f(N1, N2,….,Nn) defined for any tuple {N1, N2,….,Nn} Expectation value (mean) Gauss’ law of error propagation: Functions of Stochastic Variables Statistics Further terms if Ni not independent ( correlations)Otherwise, individual component variances (Df)2 add.

15. Example: Spectral Analysis Peak Area A Background B B1 B2 Adding or subtracting 2 Poisson distributed numbers N1 and N2:Variances always add Analyze peak in range channels c1 – c2: beginning of background left and right of peak n = c1 – c2 +1.Total area = N12 =A+B N(c1)=B1, N(c2)=B2, Linear background <B> = n(B1+B2)/2 Peak Area A: Statistics

16. Confidence Level Assume normally distributed observable x: Measured Probability Sample distribution with data set  observed average <x> and std. error s approximate population. Confidence level CL (Central Confidence Interval): Statistics With confidence level CL (probability in %), the true value <xpop> differs by less than d = ns frommeasured average. Trustworthy exptl. results quote ±3s error bars!

17. Example: Search for rare decay with decay rate l, observe no counts within time Dt. Decay probability law dP/dt=-dN/Ndt= exp {- l·t}. P(l,t) = symmetric in l and t Setting Confidence Limits no counts in Dt normalized P normalized P Upper limit Statistics Higher confidence levels CL (0  CL  1) larger upper limits for a given time Dt inspected. Reduce limit by measuring for longer period.

18. Measurement of correlations between observables y and x: {xi,yi| i=1-N} Hypothesis: y(x) =f(c1,…,cm; x). Parameters defining f: {c1,…,cm}ndof=N-m degrees of freedom for a “fit” of the data with f. Maximum Likelihood for every data point Maximize simultaneous probability Statistics When is c2 as good as can be? Minimize chi-squared by varying {c1,…,cm}: ∂c2/∂ci = 0

19. Minimizing c2 Example: linear fit f(a,b;x) = a + b·x to data set {xi, yi, si} Minimize: Equivalent to solving system of linear equations Statistics

20. Distribution of Chi-Squareds <c2>ndof=5 u:=c2 Distribution of possible c2for data sets distributed normally about a theoretical expectation (function) with ndof degrees of freedom: (Stirling’s formula) Reduced c2: Statistics Should be P  0.5 for a reasonable fit

21. CL for c2-Distributions Statistics 1-CL

22. Correlations in Data Sets uncorrel. P(x,y) y y x x a Correlations within data set. Example: yi small whenever xi small correlated P(x,y) Statistics

23. Correlations in Data Sets uncorrelated c2 surface cj cj ci ci correlated c2 surface uncertainties of deduced most likely parameters ci (e.g., a, b for linear fit) depend on depth/shallowness and shape of the c2 surface Statistics

24. Multivariate Correlations Smoothed c2surface cj initial guess search path ci • Different search strategies:Steepest gradient, Newton method w or w/odamping of oscillations,Biased MC:Metropolis MC algorithms,Simulated annealing (MC derived from metallurgy), • Various software packagesLINFIT, MINUIT,…. Statistics